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T H E 0 R I A
MOTVS CORPORVM
COELESTIYM
IN
SECTIONIBVS CONICIS SOLEM AMBIENTIVM
A V C T O R E
CAROLO FRIDERICO GAVSS.
THEORY
MOTION OF THE HEAVENLY BODIES MOVING ABOUT
THE SUN IN CONIC SECTIONS:
A TBAN8LATION OP
GAUSS'S "THEORIA MOTUS."
WITH AN APPENDIX.
CHARLES HENRY DAVIS,
COMMANDER UNITED STATES NAVY, SUPEKINTBNDENT OP THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC.
BOSTON:
LITTLE, BROWN AND COMPANY.
185 7.
Published under the Authority of the Navy Department by the Nautical Almanac and
Smithsonian Institution.
TRANSLATOE'S PREFACE.
In 1852, a pamphlet, entitled The Computation of an Orbit from Three Complete
Observations, was published, under the authority of the Navy Department, for the use
of the American Ephemeris and Nautical Almanac, the object of which was to excerpt
from various parts of Gauss's Theoria Motus, and to arrange in proper order the numer-
ous details which combine to form this complicated problem. To these were added an
Appendix containing the results of Professor Eistcke's investigations, Ueber den Ausnah-
mefall einer doppelten Bahnbestimmung aus denselben drei geocentrischen Oertern (Ab-
handlungen der Akademie der Wissenschaften zu Berlin, 1848), and also Professor Peirce's
Graphic Delineations of the Curves showing geometrically the roots of Gauss's Equa-
tion IV. Article 141.
After this pamphlet was completed, the opinion was expressed by scientific friends
that a complete translation of the Theoria Motus should be undertaken, not only to meet
the wants of the American Ephemeris, but those also of Astronomers generally, to whom
this work (now become very rare and costly) is a standard and permanent authority.
This undertaking has been particularly encouraged by the Smithsonian Institution,
which has signified its high estimate of the importance of the work, by contributing to
its publication. And by the authority of Hon. J. C. Dobbin, Secretary of the Navy, this
Translation is printed by the joint contributions of the Nautical Almanac and the Smith-
sonian Institution.
The notation of Gauss has been strictly adhered to throughout, and the translation
has been made as nearly literal as possible. No pains have been spared to secure typo-
graphical accuracy. All the errata that have been noticed in Zach's Monatliche Corre-
spondenz, the Berliner Astronomisches Jahrbuch, and the Astronomische Nachrichten, have
(V)
VI TRANSLATOR'S PREFACE.
been corrected, and in addition to these a considerable number, a list of which will be
found in Gould's Astronomical Journal, that were discovered by Professor Chadve:n'et
of the United States Naval Academy, who has examined the formulas of the body of
the work with great care, not only by comparison with the original, but by independent
verification. The proof-sheets have also been carefully read by Professor Phillips, of
Chapel Hill, North Carolina, and by Mr, Runkle and Professor Winlock of the Nautical
Almanac office.
The Appendix contains the results of the investigations of Professor Encke and
Professor Peirce, from the Appendix of the pamphlet above referred to, and other mat-
ters which, it is hoped, will be found interesting and useful to the practical computer,
among which are several valuable tables : A Table for the Motion in a Parabola from
LeVerrier's Annates de U Observatoire Imperial de Paris, Bessel's and Posselt's
Tables for Ellipses and Hyperbolas closely resembling the Parabola, and a convenient
Table by Professor Hubbard for facilitating the use of Gauss's formulas for Ellipses and
Hyperbolas of which the eccentricities are nearly equal to unity. And in the form of
notes on their appropriate articles, useful formulas by Bessel, Nicolai, Encke, Gauss,
and Peirce, and a summary of the formulas for computing the orbit of a Comet,
with the accompanying Table, from Olbers's Abhandlung ueber die leichteste und be-
quemste Methode die Balm eines Cometen zu berechnen. "Weimar, 1847.
CONTENTS
Page
Pkeface ix
FIKST BOOK.
GENERAL RELATIONS BETWEEN THE QUANTITIES BY WHICH THE MOTIONS
OF HEAVENLY BODIES ABOUT THE SUN ARE DEFINED.
First Section. — Relations pertaining simply to position in the Orbit 1
Second Section. — Relations pertaining simply to Position in Space . . . . • 54
Third Section. — Relations between Several Places in Orbit ....... 100
Fourth Section. — Relations between Several Places in Space 153
SECOND BOOK.
INVESTIGATION OF THE ORBITS OF HEAVENLY BODIES FROM GEOCENTRIC
OBSERVATIONS.
First Section. — Determination of an Orbit from Three Complete Observations . . .161
Second Section. — Determination of an Orbit from Fom- Observations, of which Two only
are Complete 234
Third Section. — Determination of an Orbit satisfying as nearly aS possible any number of
Observations whatever 2^9
Fourth Section. — On the Determination of Orbits, taking into accomit the Perturbations . 274
Appendix 279
Tables 329
(vii)
CAMBRIDGE :
PEINTED BT ALLEN AND FARNHi
PREFACE.
After the laws of planetary motion were discovered, the genius of Kepler
was not without resources for deriving from observations the elements of mo-
tion of individual 'planets. Tycho Brake, by whom practical astronomy had
been carried to a degree of perfection before imknown, had observed all the
planets through a long series of years with the greatest care, and with so
much perseverance, that there remained to Kepler, the most worthy inheritor
of such a repository, the trouble only of selecting what might seem suited
to any special purpose. The mean motions of the planets already deter-
mined with great precision by means of very ancient observations diminished
not a little this labor.
Astronomers who, subsequently to Kepler, endeavored to determine still
more accurately the orbits of the planets with the aid of more recent or
better observations, enjoyed the same or even greater facilities. For the
problem was no longer to deduce elements wholly unknown, but only
slightly to correct those already known, and to define them within narrower
limits.
The principle of universal gravitation discovered by the illustrious Newton
X PREFACE.
opened a field entirely new, and showed that all the heavenly bodJes, at
least those the motions of which are regulated by the attraction of the sun,
must necessarily, conform to the same laws, with a slight modification only,
by which Ej:pler had found the five planets to be governed. Kepler, rely-
ing upon the evidence of observations, had announced that the orbit of every
planet is an ellipse, in which the areas are described uniformly about the
sun occupying one focus of the ellipse, and in such a manner that in differ-
ent ellipses the times of revolution are in the sesquialteral ratio of the semi-
axes-major. On the other hand, Newton, starting from the principle of
universal gravitation, demonstrated d, priori that all bodies controlled by the
attractive force of the sun must move in conic sections, of which the planets
present one form to us, namely, ellipses, while the remaining forms, parabo-
las and hyperbolas, must be regarded as being equally possible, provided
there may be bodies encountering the force of the sun with the requisite
velocity; that the sun must always occupy one focus of the conic section;
that the areas which the same body describes in different times about the
sun are proportional to those times; and finally, that the areas described
about the sun by different bodies, in equal times, are in the subduplicate
ratio of the semiparameters of the orbits: the latter of these laws, identical
in elliptic motion with the last law of Kepler, extends to the parabolic and
hyperbolic motion, to which Kepler's law cannot be applied, because the rev-
olutions are wanting. The clue was now discovered by following which it
became possible to enter the hitherto inaccessible labyrinth of the motions of
the comets. And this was so successful that the single hjqsothesis, that their
orbits were parabolas, sufficed to explain the motions of all the comets which
had been accurately observed. Thus the system of universal gravitation had
PREFACE. Xi
paved' the way to new and most brilliant triumphs in analysis j and the
comets, up to that time wholly unmanageable, or soon breaking from the
restraints to which they seemed to be subjected, having now submitted to
control, and being transformed from enemies to guests, moved on in the
paths marked out by the calculus, scrupulously conforming to the same eter-
nal laws that govern the planets.
In determining the parabolic orbits of comets from observation, difficul-
ties arose far greater than in determining the elliptic orbits of planets, and
principally from this source, that comets, seen for a brief interval, did not
afford a choice of observations particularly suited to a given object : but the
geometer was compelled to employ those which happened to be furnished
him, so that it became necessary to make use of special methods seldom
applied in planetary calculations. The great Newton himself, the first geome-
ter of his age, did not disguise the difficulty of the problem: as might have
been expected, he came out of this contest also the victor. Since the time
of Newton, many geometers have labored zealously on the same problem,
with various success, of course, but still in such a manner as to leave but
little to be desired at the present time.
The truth, however, is not to be overlooked that in this problem the
difficulty is very fortunately lessened by the knowledge of one element of
the conic section, since the major-axis is put equal to infinity by the very
assumption of the parabolic orbit. For, all parabolas, if position is neg-
lected, differ among themselves only by the greater or less distance of the
vertex from the focus; while conic sections, generally considered, admit of
infinitely greater variety. There existed, in point of fact, no sufficient reason
why it should be taken for granted that the paths of comets are exactly
xii PREFACE.
parabolic: on the contrary, it must be regarded as in the highest degree
improbable that nature should ever have favored such an hypothesis. Since,
nevertheless, it was known, that the phenomena of a heavenly body moving
in an ellipse or hyperbola, the major-axis of which is very great relatively to
the parameter, differs very little near the perihelion from the motion in a
parabola of which the vertex is at the same distance from the focus; and
that this difference becomes the more inconsiderable the greater the ratio of
the axis to the parameter : and since, moreover, experience had shown that
between the observed motion and the motion computed in the parabolic
orbit, there remained differences scarcely ever greater than those which might
safely be attributed to errors of observation (errors quite considerable in
most cases) : astronomers have thought proper to retain the parabola, and
very properly, because there are no means whatever of ascertaining satis-
factorily what, if any, are the differences from a parabola. We must except
the celebrated comet of Halley, which, describing a very elongated ellipse and
frequently observed at its return to the perihelion, revealed to us its periodic
time ; but then the major-axis being thus known, the computation of the re-
maining elements is to be considered as hardly more difficult than the determi-
nation of the parabolic orbit. And we must not omit to mention that astrono-
mers, in the case of some other comets observed for a somewhat longer time,
have attempted to determine the deviation from a parabola. However, all
the methods either proposed or used for this object, rest upon the assumption
that the variation from a parabola is inconsiderable, and hence in the trials
referred to, the parabola itself, previously computed, furnished an approximate
idea of the several elements (except the major-axis, or the time of revolu-
tion depending on it), to be corrected by only slight changes. Besides, it
PREFACE. XIU
must be acknowledged, that the whole of these trials hardly served in any
case to settle any thing with certainty, if, perhaps, the comet of the year
1770 is excepted.
As soon as it was ascertained that the motion of the new planet, discov-
ered in 1781, could not be reconciled with the parabolic hypothesis, astrono-
mers undertook to adapt a circular orbit to it, which is a matter of simple
and very easy calculation. By a happy accident the orbit of this planet had
but a small eccentricity, in consequence of which the elements resulting from
the circular hypothesis sufficed at least for an approximation on which could
be based the determination of the elliptic elements. There was a concur-
rence of several other very favorable circumstances. For, the slow motion of
the planet, and the very small inclination of the orbit to the plane of the
ecliptic, not only rendered the calculations much more simple, and allowed
the use of special methods not suited to other cases; but they removed the
apprehension, lest the planet, lost in the rays of the sun, should subsequently
elude the search of observers, (an apprehension which some astronomers might
have felt, especially if its light had been less brilliant) ; so that the more
accurate determination of the orbit might be safely deferred, until a selection
could be made from observations more frequent and more remote, such as
seemed best fitted for the end in view.
Thus, in every case in which it was necessary to deduce the orbits of
heavenly bodies from observations, there existed advantages not to be de-
spised, suggesting, or at any rate permitting, the application of special
methods; of which advantages the chief one was, that by means of hypo-
thetical assumptions an approximate knowledge of some elements could be
XIV PREFACE.
obtained before the computation of the elliptic elements was commenced.
Notwithstanding this, it seems somewhat strange that the general problem, —
To determine the orhit of a heavenly hodt/, ivithout any hypothetical assumption,
from observations not embracing a great period ,of time, and not alloimig a selection
tvith a view to the application of special methods, was almost wholly neglected up
to the beginning of the present century j or, at least, not treated by any one
in a manner worthy of its importance; since it assuredly commended itself
to mathematicians by its difficulty and elegance, even if its great utility in
practice were not apparent. An opinion had universally prevailed that a
complete determination from observations embracing a short interval of time
was impossible, — an ill-founded opinion, — for it is now clearly shown that
the orbit of a heavenly body may be determined quite nearly from good
observations embracing only a few days; and this without any hypothetical
assumption.
Some ideas occurred to me in the month of September of the year 1801,
engaged at the time on a very different subject, which seemed to point to
the solution of the great problem of which I have spoken. Under such cir-
cumstances we not unfrequently, for fear of being too much led away by
an attractive investigation, suffer the associations of ideas, which, more atten-
tively considered, might have proved most fruitful in results, to be lost from
neglect. And the same fate might have befallen these conceptions, had they
not happily occurred at the most propitious moment for their preservation
and encouragement that could have been selected. For just about this time
the report of the new planet, discovered on the first day of January of that
year with the telescope at Palermo, was the subject of universal conversation;
PREFACE. XV
and soon afterwards the observations made by that distinguished astronomer
PiAZZi from the above date to the eleventh of February were pubHshed. No-
where in the annals of astronomy do we meet with so great an opportunity,
and a greater one could hardly be imagined, for showing most strikingly, the
value of this problem, than in this crisis and urgent necessity, when all hope
of discovering in the heavens this planetary atom, among innumerable small
stars after the lapse of nearly a year, rested solely upon a sufficiently ap-
proximate knowledge of its orbit to be based upon these very few observa-
tions. Could I ever have found a more seasonable opportunity to test the
practical value of my conceptions, than now in employing them for the de-
termination of the orbit of the planet Ceres, which during these forty-one
days had described a geocentric arc of only three degrees, and after the
lapse of a year must be looked for in a region of the heavens very remote
from that in which it was last seen ? This first application of the method
was made in the month of October, 1801, and the first clear night, when
the planet was sought for* as directed by the numbers deduced from it, re-
stored the fugitive to observation. Three other new planets, subsequently
discovered, furnished new opportunities for examining and verifying the efii-
ciency and generality of the method.
Several astronomers wished me to publish the methods employed in these
calculations immediately after the second discovery of Ceres ; but many
things — other occupations, the desire of treating the subject more fully at
some subsequent period, and, especially, the hope that a further prosecution
of this investigation would raise various parts of the solution to a greater
By de Zach, December 7, 1801.
2
Xvi PREFACE.
degree of generality, simplicity, and elegance, — prevented my complying at
the time with these friendly solicitations. I was not disappointed in this ex-
pectation, and have no cause to regret the delay. For, the methods first
employed have undergone so many and such great changes, that scarcely
any trace of resemblance remains between the method in which the orbit of
Ceres was first computed, and the form given in this work. Although it
would be foreign to my purpose, to narrate in detail all the steps by
which these investigations have been gradually perfected, still, in several
instances, particularly when the problem was one of more importance than
usual, I have thought that the earlier methods ought not to be wholly sup-
pressed. But in this work, besides the solutions of the principal problems,
I have given many things which, during the long time I have been en-
gaged upon the motions of the heavenly bodies in conic sections, struck
me as worthy of attention, either on account of their analytical elegance,
or more especially on account of their practical utility. But in every case
I have devoted greater care both to the subjects and methods which are
peculiar to myself, touching lightly and so far only as the connection seemed
to require, on those previously known.
The whole work is divided into two parts. In the First Book are de-
veloped the relations between the quantities on which the motion of the
heavenly bodies about the sun, according to the laws of Keplek, depends;
the two first sections comprise those relations in which one place only is
considered, and the third and fourth sections those in which the relations
between several places are considered. The two latter contain an explanation
of the common methods, and also, and more particularly, of other methods,
greatly preferable to them in practice if I am not mistaken, by means of
PREFACE. Xvii
which we pass from the known elements to the phenomena; the former treat
of many most important problems which prepare the way to inverse pro-
cesses. Since these very phenomena result from a certain artificial and intri-
cate complication of the elements, the nature of this texture must be thor-
oughly examined before we can undertake with hope of success to disentangle
the threads and to resolve the fabric into its constituent parts. Accordingly,
in the First Book, the means and appliances are provided, by means of which,
in ,the second, this difficult task is accomplished ; the chief part of the labor,
therefore, consists in this, that these means should be properly collected to-
gether, should be suitably arranged, and directed to the proposed end.
The more important problems are, for the most part, illustrated by appro-
priate examples, taken, wherever it was possible, from actual observations.
In this way not only is the efl&cacy of the methods more fully established
and their use more clearly shown, but also, care, I hope, has been taken that
inexperienced computers should not be deterred from the study of these sub-
jects, which undoubtedly constitute the richest and most attractive part of
theoretical astronomy.
GoTTiNGEN, March. 28, 1809.
FIRST BOOK.
GENERAL RELATIONS BETWEEN THOSE QUANTITIES BY WHICH THE
MOTIONS OF HEAVENLY BODIES ABOUT THE SUN ARE DEFINED.
FIRST SECTION.
KELATIONS PERTAINING SIMPLY TO POSITION IN THE ORBIT.
1.
In this work we shall consider the motions of the heavenly bodies so far only
as they are controlled by the attractive force of the sun. All the secondary
planets are therefore excluded from our plan, the perturbations which the
primary planets exert upon each other are excluded, as is also all motion of
rotation. "We regard the moving bodies themselves as mathematical points, and
we assume that all motions are performed in obedience to the following laws,
which are to be received as the basis of all discussion in this work.
I. The motion of every heavenly body takes place in the same fixed
plane in which the centre of the sun is situated.
II. The path described by a body is a conic section having its focus in the
centre of the sun.
III. The motion in this path is such that the areas of the spaces described
about the sun in different intervals of time are proportional to those intervals.
Accordingly, if the times and spaces are expressed in numbers, any space what-
ever divided by the time in which it is described gives a constant quotient.
1
2 RELATIONS PERTAINING SBITLY [BoOK I.
IV. For different bodies moving about the sun, the squares of these quotients
are in the compound ratio of the parameters of their orbits, and of the sum of the
masses of the sun and the moving bodies.
Denoting, therefore, the parameter of the orbit in which the body moves by
2ji9, the mass of this body by f^ (the mass of the sun being put =1), the area it
describes about the sun in the time t by ^(/, then ^ , ^q j_ n will be a constant
for all heavenly bodies. Since then it is of no importance which body we use
for determining this number, we will derive it from the motion of the earth, the
mean distance of which from the sun we shall adopt for the unit of distance ; the
mean solar day will always be our unit of time. Denoting, moreover, by tt the
ratio of the circumference of the circle to the diameter, the area of the entire
ellipse described by the earth will evidently be n \J p, which must therefore be
put = ^^, if by ^ is understood the sidereal year; whence, our constant becomes
— — — — - . In order to ascertain the numerical value of this constant, here-
after to be denoted by k, we will put, according to the latest determination, the
sidereal year or if = 365.2563835, the mass of the earth, or fi:^ =:
0.0000028192, whence results
log27r 0.7981798684
Compl. log^ 7.4374021852
Compl. log. v/(14-ii^) . . . 9.9999993878
log /f 8.2355814414
k= 0.01720209895.
2.
The laws above stated differ from those discovered by our own Kepler
in no other respect than this, that they are given in a form applicable to all kinds
of conic sections, and that the action of the moving body on the sun, on which
depends the factor y'(l-)-^), is taken into account. If we regard these laws as
phenomena derived from innumerable and indubitable observations, geometry
shows what action ought in consequence to be exerted upon bodies moving about
Sect. 1.] to position in the orbit. 3
the sun, in order that these phenomena may be continually produced. In this
way it is found that the action of the sun upon the bodies moving about it is
exerted just as if an attractive force, the intensity of which is reciprocally
proportional to the square of the distance, should urge the bodies towards the
centre of the sun. If now, on the other hand, we set out with the assumption of
such an attractive force, the phenomena are deduced from it as necessary
consequences. It is sufficient here merely to have recited these laws, the con-
nection of which with the principle of gravitation it will be the less necessary to
dwell upon in this place, since several authors subsequently to the eminent
Newton have treated this subject, and among them the illustrious La Place, in
that most perfect work the Mecanique Celeste, in such a manner as to leave
nothing further to be desired.
3.
Inquiries into the motions of the heavenly bodies, so far as they take place in
conic sections, by no means demand a complete theory of this class of curves ;
but a single general equation rather, on which all others can be based, will answer
our purpose. And it appears to be particularly advantageous to select that one
to which, while investigating the curve described according to the law of attrac-
tion, we are conducted as a characteristic equation. If we determine any place
of a body in its orbit by the distances x, y, from two right lines drawn in the
plane of the orbit intersecting each other at right angles in the centre of the
sun, that is, in one of the foci of the curve, and further, if we denote the distance
of the body from the sun by r (always positive), we shall have between r, x, p,
the linear equation r-[- a :«;-)- /3^=r/, in which «, (3, / represent constant quan-
tities, / being from the nature of the case always positive. By changing the
position of the right lines to which x,i/, are referred, this position being essentially
arbitrary, provided only the lines continue to intersect each other at right angles,
the form of the equation and also the value of / will not be changed, but the
values of a and /i will vary, and it is plain that the position may be so determined
that (i shall become = 0, and «, at least, not negative. In this way by putting for
«, ?'; respectively e.p, our equation takes the form r-\-ex=^p. The right line to
4 RELATIONS PERTAINING SBIPLY BoOK I.
which the distances y are referred in this case, is called the line of ajysides, p is the
semi-parameter, e the eccentriciti/ ; finally the conic section is distinguished by the
name of ellijjse, parabola, or hyperlola, according as e is less than unity, equal to
unity, or greater than unity.
It is readily perceived that the position of the line of apsides would be
fully determined by the conditions mentioned, with the exception of the single
case where both a and /5 were = 0 ; in which case r is always =^?, whatever the
right lines to which x, y, are referred. Accordingly, since we have 6 = 0, the
curve (which will be a circle) is according to our definition to be assigned to
the class of ellipses, but it has this peculiarity, that the position of the apsides
remains wholly arbitrary, if indeed we choose to extend that idea to such a case.
4.
Instead of the distance x let us introduce the angle v, contained between the
line of apsides and a straight line drawn from the sun to the place of the body
{the radius vector), and this angle may commence at that part of the line of apsides
at which the distances x are positive, and may be supposed to increase in the
direction of the motion of the body. In this way we have x = r cos v, and thus
our formula becomes r = :r-r , from which immediately result the following
1 -|- e cos w ' -^ °
conclusions : —
I. For v^ 0, the value of the radius vector r becomes a minimum, that is,
= l^T^ • this j)oint is called the perihelion.
II. For opposite values of v, there are corresponding equal values of r ; con-
sequently the line of apsides divides the conic section into two equal parts.
III. In the ellipse, v increases continuously from y = 0, until it attains its
maximum value, y^^, in a^M?'ra, corresponding to y = 180°; after aphelion, it
decreases in the same manner as it had increased, until it reaches the perihelion,
corresponding to v:=^ 360°. That portion of the line of apsides terminated at one
extremity by the perihelion and at the other by the aphelion is called the major
Sect. 1.] to position in the orbit. 5
axis ; hence the semi-axis major, called also the mean distance, = y^ — ; the dis-
tance of the middle point of the axis {the centre of the .ellipse) from the focus will
be ^ = ea, denoting by a the semi-axis major.
IV. On the other hand, the aphelion in its proper sense is wanting in the
parabola, but r is increased indefinitely as v approaches -|- 180°, or — 180°. For
V = + 180° the value of r becomes infinite, which shows that the curve is not cut
by the line of apsides at a point opposite the perihelion. Wherefore, we cannot,
with strict propriety of language, speak of the major axis or of the centre of the
curve ; but by an extension of the formulas found in the ellipse, according to the
established usage of analysis, an infinite value is assigned to the major axis, and
the centre of the curve is placed at an infinite distance from the focus.
Y. In the hyperbola, lastly, v is confined within still narrower limits, in fact
between v= — (180° — \\i), and y = -|-(180° — ^i), denoting by x^ the angle of
which the cosine =-. For whilst v approaches these limits, r increases to
infinity ; if, in fact, one of these two limits should be taken for v, the value of r
would result infinite, which shows that the hyperbola is not cut at all by a right
line inclined to the line of apsides above or below by an angle 180° — if. For
the values thus excluded, that is to say, from 180° — i// to 180°-[-^j our formula
assigns to r a negative value. The right line inclined by such an angle to the
line of apsides does not indeed cut the hyperbola, but if produced reversely,
meets the other branch of the hyperbola, which, as is known, is wholly separ
rated from the first branch and is convex towards that focus, in which the sun is
situated. But in our investigation, which, as we have already said, rests upon the
assumption that r is taken positive, we shall pay no regard to that other branch
of the hyperbola in which no heavenly body could move, except one on which
the sun should, according to the same laws, exert not an attractive but a repulsive
force. Accordingly, the aphelion does not exist, properly speaking, in the hyper-
bola also ; that point of the reverse branch which lies in the line of apsides,
and which corresponds to the values y = 180°, r = — ^~ii might be consid-
ered as analogous to the aphelion. If now, we choose after the manner of the
6 KELATIONS PERTAINING SBIPLY [BoOK I.
ellipse to call the value of the expression _^ , even here where it becomes
negative, the semi-axis major of the hyperbola, then this quantity indicates
the distance of the point just mentioned from the perihelion, and at the
same time the position opposite to that which occurs in the ellipse. In the
same way ^ , that is, the distance from the focus to the middle point between
these two points (the centre of the hyperbola), here obtains a negative value on
account of its opposite direction.
We call the angle v the true anomaly of the moving body, which, in the
parabola is confined within the limits — 180° and -{- 180°, in the hyperbola
between — (180° — i/^) and -|- (180° — t/^), but which in the ellipse runs through
the whole circle in periods constantly renewed. Hitherto, the greater number of
astronomers have been accustomed to count the true anomaly in the ellipse not
from the perihelion but from the aphelion, contrary to the analogy of the parabola
and hyperbola, where, as the aphelion is wanting, it is necessary to begin from the
perihelion : we have the less hesitation in restoring the analogy among all classes
of conic sections, that the most recent French astronomers have by their example
led the way.
It is frequently expedient to change a little the form of the expression
r = r—r-^ : the following forms will be especially observed : —
1 -|- e cos v' " r ./
y— P P
1 _[- e — 2 e sin^ ^v 1 — e-\-2e cos^ ^ v
^_ P
(1 -|- e) cos^ \v -\-{\ — e) sin'-^ ^ v '
Accordingly, we have in the parabola
P ^
^~2cos2it;'
in the hyperbola the following expression is particularly convenient,
^_ -Pcost/; ^
2cos^(i>-l-i/;)cos^(u — i/))*
kJiiCT. 1.]
TO POSITION IN THE ORBIT.
6.
Let us proceed now to the comparison of the motion with the time. Putting,
as in Art. 1, the space described about the sun in the time t=hg, the mass of the
moving body = ^, tliat of the sun being taken = 1, we have g =zJi:t\^p\j {l-\- ^a).
The differential of the space = hrrdv, from which there results Jtt\JpsJ {1 ^ ^)
=:frrdiV, this integral being so taken that it will vanish for ^ = 0. This integra-
tion must be treated differently for different kinds of conic sections, on which
account, we shall now consider each kind separately, beginning with the ELLIPSE.
Since r is determined from v by means of a fraction, the denominator of which
consists of two terms, we will remove this inconvenience by the introduction of a
new quantity in the place of v. For this purpose we will put tan ^v ^ y~T~ =^
tan h E, by which the last formula for r in the preceding article gives
^ — (l^e)cos^v—^\l-\-e^l—e} — l-ee^^ 6 COS ^ j .
Moreover we have — rj-m = — ¥t-\/ t-t—) and consequently dv= —M. r:
hence
'■'■d^=v-^=^i(i-"-^)<'^'
and integrating,
Mslp^{l-^lJi)= P^ 3(^— g sin ^)-f Constant.
Accordingly, if we place the beginning of the time at the perihelion passage, where
^) = 0, ^= 0, and thus constant = 0, we shall have, by reason of ^_^^ ■= a,
J
In this equation the auxiliary angle E, which is called the eccentric anomaly^
must be expressed in parts of the radius. This angle, however, may be retained
in degrees, etc., if e sin E and — ^ 3 are also expressed in the same manner ;
these quantities will be expressed in seconds of arc if they are multiplied by the
8 KELATIONS PERTAINING SBIPLY [BoOK I.
number 206264.81. We can dispense with the multiphcation by the last quan-
tity, if we employ directly the quantity h expressed in seconds, and thus put,
instead of the value before given, Ti = 3548".18761, of which the logarithm =
3.5500065746. The quantity — 3 expressed in this manner is called the
mean anomaly, which therefore increases in the ratio of the time, and indeed every
day by the increment , , called the mean daily motion. "We shall denote
the mean anomaly by M.
7:
Thus, then, at the perihelion, the true anomaly, the eccentric anomaly, and the
mean anomaly are = 0 ; after that, the true anomaly increasing, the eccentric
and mean are augmented also, but in such a way that the eccentric continues to
be less than the true, and the mean less than the eccentric up to the aphelion,
where all three become at the same time = 180°; but from this point to
the perihelion, the eccentric is always greater than the true, and the mean
greater than the eccentric, until in the perihelion all three become = 360°, or,
which amounts to the same thing, all are again = 0. And, in general, it is
evident that if the eccentric U and the mean M answer to the true anomaly v,
then the eccentric 360° — U and the mean 360° — M correspond to the true
360° — V. The difference between the true and mean anomalies, v — J^ is called
the equation of the centre, which, consequently, is positive from the perihelion
to the aphelion, is negative from the aphelion to the perihelion, and at the
perihelion and aphelion vanishes. Since, therefore, v and 31 run through an
entire circle from 0 to 360° in the same time, the time of a single revolution,
which is also called the periodic time, is obtained, expressed in days, by dividing
360° by the mean daily motion v ^ 7~ , from which it is apparent, that for dif
a-
ferent bodies revolving about the sun, the squares of the periodic times are pro-
portional to the cubes of the mean distances, so far as the masses of the bodies,
or rather the inequality of their masses, can be neglected.
Sect. 1.] to position in the orbit. 9
8.
Let us now collect together those relations between the anomalies and the
radius vector which deserve particular attention, the derivation of which will
present no difficulties to any one moderately skilled in trigonometrical analysis.
Greater elegance is attained in most of these formulas by introducing in the
place of e the angle the sine of which = e. This angle being denoted by 9, we
have
y/(l_gg) = cos9), y/(l-[-e)=:cos(45° — i9))v/2,
^(1 — e) = cos(45°+f9)v/2, y/l^==tan(45° — ^9),
y/ (1 _f_ e) _^ ^(1 _ e) = 2 cos 1 9, sj {\^ e) — ^{\ — e) = 'lmx\<^.
The following are the principal relations between a, p, r, e, cp, v, E, M.
I. p = a cos^ (p
1 -j- e cos «
III. r = a{l — ecosE)
l\r ^^ E7 cosv-4-e cos^ — e
IV. cos^ = --| — =c_ or cos ZJ = :; ^
1 -f- e cos w ' 1 — e cos Mi
V. smiE=J^(l — cosE)=smivJy-^~^^-^
' ^ ^ V 1 -|- e cos s;
= Sin i . ^^^^^ ^ sin ^ . y/^^^
VL cos^^=:v/^l + cos^)=cos^.y/j-li^^
= cos ^y 1/— ^-i^ = COS hvJ—rr- — ^
Vn. tani^=tan^z;tan(45°— ^9)
VIII. sin ^^"^"""'""^^ '"''"''
p a cos qo
IX. rcosz; = a(cos^— e) = 2«cos(^^+i9) + 45°)cos(^J^— ig) — 45°)
X. sin h{v — ^) = sin J 9 sin ^; w - = sin J g) sin ^ i/ -
XL sin i (z; -}- -^) = cos i 9 sin tJ i/ - =: cos J 9 sin J' w-
XIL M=E—esmE.
2
10 RELATIONS PERTAINING SIMPLY [BoOK I.
If a perpendicular let fall from any point whatever of the ellipse upon the
line of apsides is extended in the opposite direction until it meets the circle
described with the radius a about the centre of the ellipse, then the inclination to
the line of apsides of that radius which corresponds to the point of intersection
(understood in the same way as above, in the case of the true anomaly), will
be equal to the eccentric anomaly, as is inferred without difficulty from equation
IX. of the preceding article. Further, it is evident that r sin v is the distance of
any point of the ellipse from the line of apsides, which, since by equation VIII. it
= a cos 9 sin E, will be greatest for E^=. 90°, that is in the centre of the ellipse.
This greatest distance, which = a cos y ^=^ -^ ■=.'^ap, is called the semi-axis minor.
In the focus of the ellipse, that is for y = 90°, this distance is evidently ■=p, or
equal the semi-parameter.
10.
The equations of article 8 comprise all that is requisite for the computation
of the eccentric and mean anomalies from the true, or of the eccentric and true
from the mean. Formula VII. is commonly employed for deriving the eccentric
from the true ; nevertheless it is for the most part preferable to make use of
equation X. for this purpose, especially when the eccentricity is not too great, in
which case E can be computed with greater accuracy by means of X. than of
VII. Moreover, if X. is employed, the logarithm of sine E required in XII. is
had immediately by means of VIII. : if VII. were used, it would be neces-
sary to take it out from the tables; if, therefore, this logarithm is also taken
from the tables in the latter method, a proof is at once obtained that the calcula-
tion has been correctly made. Tests and proofs of this sort are always to be
highly valued, and therefore it will be an object of constant attention with us to
provide for them in all the methods delivered in this work, where indeed it can
be conveniently done. We annex an example completely calculated as a more
perfect illustration.
Sect. 1.] to position in the orbit. 11
Given V = 310° 55' 29^64, 9 = 14° 12' r.87, log r = 0.3307640 ; p, a, U, M,
are required.
log sin 9 .... 9.3897262
log cosy .... 9.8162877
9.2060139 whence e cos y = 0.1606993
log (1 -I- e cos?;). . 0.0647197
logr 0.3307640
\ogp 0.3954837
logcos> .... 9.9730448
log a 0.4224389
log sine; .... 9.8782740 ;2*
logi/f .... 0.0323598.5
9.8459141.5W
log sin ^ 9 ... 9.0920395
log sin ^(y — ^) . 8.9379536.5^, hence ^(y — ^) = — 4° 58' 22".94;
y — ^ = — 9° 56' 45".88 ; E= 320° 52' 15".52.
Further, we have
■ „ „^„w^^^ Calculation of log sin E by formula Vm.
loge . . . . 9.3897262 ^
1 oAcoc.( o KQT^/ion log-smy .... 9.8135543w
log 206264.8 . 5.3144251 ^ p
log e in seconds 4.7041513 log cos 9 9.9865224
logsin^. . . 9.8000767w logsin^ 9.8000767w
4.5042280 n, hence e sin ^ in seconds = 31932".14 = 8° 52'
12".14 ; and M= 329° 44' 27".66.
The computation of E by formula VII. would be as follows : —
^?; = 155°27'44".82 log tan ^y .... 9.6594579?2
] 45° — l9) = 37"53'59".065 log tan (45° — i 9) . 9.8912427
log tan ^^ . . . . 9.5507006 «
whence hE= 160''26'7".76, and E= 320°52'15".52, as above.
* The letter « affixed to a logarithm signifies that the number corresponding to it is negative.
12 RELATIONS PERTAINING SBIPLY [BoOK I.
11.
The inverse problem, celebrated under the title of Kepler's problem, that of
finding the true anomaly and the radius vector from the mean anomaly, is much
more frequently used. Astronomers are in the habit of putting the equation of
the centre in the form of an infinite series proceeding according to the sines of the
angles M, 2M, SM, etc., each one of the coefficients of these sines being a series
extending to infinity according to the powers of the eccentricity. We have con-
sidered it the less necessary to dwell upon this formula for the equation of the
centre, which several authors have developed, because, in our opinion, it is by
no means so well suited to practical use, especially should the eccentricity not be
very small, as the indirect method, which, therefore, we will explain somewhat
more at leng-th in that form which appears to us most convenient.
Equation XII., U = M-\- e sin H, which is to be referred to the class of tran-
scendental equations, and admits of no solution by means of direct and complete
methods, must be solved by trial, beginning with any approximate value of jC, which
is corrected by suitable methods repeated often enough to satisfy the preceding
equation, that is, either with all the accuracy the tables of sines admit, or at least
with sufficient accuracy for the end in view. If now, these corrections are intro-
duced, not at random, but according to a safe and established rule, there is scarcely
any essential distinction between such an indirect method and the solution by
series, except that in the former the first value of the unknown quantity is in a
measure arbitrary, which is rather to be considered an advantage since a value
suitably chosen allows the corrections to be made with remarkable rapidity. Let
us suppose £ to be an approximate value of U, and z expressed in seconds the cor-
rection to be added to it, of such a value as will satisfy our equation JS=t-\-x.
Let e sin e, in seconds, be computed by logarithms, and when this is done, let the
change of the log sin e for the change of 1" in e itself be taken from the tables ;
and also the variation of log e sin e for the change of a unit in the number e sin s ;
let these changes, without regard to signs, be respectively I, fi, in which it is
hardly necessary to remark that both logarithms are presumed to contain an
equal number of decimals. Now, if e approaches so near the correct value of ^
Sect. 1.] to position in the orbit. 13
that the changes of the logarithm of the sine from s to e-\-x, and the changes of
the logarithm of the number from e sin e to e sin (s -j- x), can be regarded as
uniform, we may evidently put
7 X
e sin (s -]- rr) = e sin £ + — ,
the upper sign belonging to the first and fourth quadrants, and the lower to the
second and third. Whence, since
e -\~x = M-\- e sin (e -f- x), we have x = -~ {^-\- & sin e — e),
and the correct value of ^,or
£ -|-a; = i!^f -]- ^sine+^=rj(7{f-|- esine — e),
the signs being determined by the above-mentioned condition.
Finally, it is readily perceived that we have, without regard to the signs,
li-.X^X-.e cos e, and therefore always ^^^, whence we infer that in the first and
last quadrant M-\- e sin e lies between e and e -\- x, and in the second and third,
e-^x between t and M-\-esin e, which rule dispenses with paying attention to the
signs. If the assumed value e differs too much from the truth to render the fore-
going considerations admissible, at least a much more suitable value will be found
by this method, with which the same operation can be repeated, once, or several
times if it should appear necessary. It is very apparent, that if the difierence
of the first value £ from the truth is regarded as a quantity of the first order, the
error of the new value would be referred to the second order, and if the operation
were further repeated, it would be reduced to the fourth order, the eighth order,
etc. Moreover, the less the eccentricity, the more rapidly will the successive
corrections converge.
12.
The approximate value of ^, with which to begin the calculation, will, in most
cases, be obvious enough, particularly where the problem is to be solved for
several values of M of which some have been already found. In the absence
of other helps, it is at least evident that U must fall between M and M±e, (the
eccentricity e being expressed in seconds, and the upper sign being used in the
14 RELATIONS rERTiVLNTNG SIMPLY [BoOK 1.
first and second quadrants, the lower in the third and fourth), wherefore, either
3f, or its value increased or diminished by any estimate whatever, can be taken
for the first value of E. It is hardly necessary to observe, that the first calcu-
lation, when it is commenced with a value having no pretension to accuracy, does
not require to be strictly exact, and that the smaller tables * are abundantly suffi-
cient. Moreover, for the sake of convenience, the values selected for £ should be
such that their sines can be taken from the tables without interpolation ; as, for
example, values to minutes or exact tens of seconds, according as the tables
used proceed by differences of minutes or tens of seconds. Every one will be
able to determine without assistance the modifications these precepts undergo if
the angles are expressed according to the new decimal division.
13.
Example. — Let the eccentricity be the same as in article 10. J!f=332°28'
54".77. There the log e in seconds is 4.7041513, therefore e= 50600" = 14° 3'20".
Now since E here must be less than M, let us in the first calculation put e ==: 326°,
then we have by the smaller tables
log sine 9.74756;?, Changeforr ... 19, whence a = 0.32.
log; c in seconds . . 4.70415
4.45171w;
hence esin£= 28295 = 7°5135. change of logarithm for a unit of the table which is here
MA- e sin £ 324 37 20 ^^^'^ *° ■"* ^^'^°^^^ . . . I6; whence /i=1.6.
differing from £ .... 1 22 40 = 4960". Hence,
?4| X 4960" = 1240" = 20'40".
Wherefore, the corrected value of E becomes 324° 37' 20"— 20' 40"= 324° 16'40",
with which we repeat the calculation, making use of larger tables.
log sin £ .... 9.7663058;e 2, = 29.25
loge 4.7041513
4.4704571 w /* = 147
* Such as those which the illusti-ious Lalaxde furnisher!.
Sect. 1.] to position in the orbit. 15
e sin e = — 29543'a8 = — 8° 12' 23'a8
M-\-esme .... 324 1631.59
differing from e . * . 8 .41.
X 29 25 .
This difference being multiplied by -^^3= „„g gives 2''.09, whence, finally, the
corrected value of ^ = 324°16'3r.59 — 2".09 = 324°16'29''.50, which is exact
within 0''.01.
14.
The equations of article 8 furnish several methods for deriving the true
anomaly and the radius vector from the eccentric anomaly, the best of which we
will explain.
I. By the common method v is determined by equation VII., and afterwards
r by equation II.; the example of the preceding article treated in this way
is as follows, retaining for jt? the value given in article 10.
i^=162°8'14".75 loge 9.3897262
log tan ^^. . . . 9.5082198n log cosy .... 9.8496597
log tan (45°— i 9) . 9.8912427 9.2393859
logtan^y . . . . 9.6169771 w ecosy =0.1735345
iv = 157° 30'4r.50 l^ 0.3954837
t' = 315 123.00 log (1-f-e cost'). . 0.0694959
"b^T 0.3259878.
II. The following method is shorter if several places are to be computed,
for which the constant logarithms of the quantities ^ a{l-\-e), )J a{l — e) should
be computed once for all. By equations V. and VI. we have
smiv^r^=smiJS )^ a{l-\-e)
cos i y \/ r = cos ^ -E' \/ a{l — e)
from which i v and log \J r are easily determined. It is true in general that if we
have Psin Q = A, P cos Q=zB, ^ is obtained by means of the formula tan
Q = ^, and then P by this, P = ^, or by P = ^^ : it is preferable to use
16
KELATIONS PERTAINING SIMPLY
[Book 1.
the former when sin Q is greater than cos Q ; the latter when cos Q is greater than
sin Q. Commonly, the problems in which equations of this kind occur (such as
present themselves most frequently in this work), involve the condition that P
should be a positive quantity ; in this case, the doubt whether Q should be taken
between 0 and 180°, or between 180° and 360°, is at once removed. But if such
a condition does not exist, this decision is left to our judgment.
We have in our example e = 0.2453162.
log sin i^ . . . 9.4867632 log cos ^^ . . . 9.9785434w
0.2588593
logv/a(14-e) .
Hence
log sm^v \J r .
log cos IvsJ r .
log cos iv . .
log y/«(1 — e)
0.1501020.
9.7456225
0.1286454
9.9656515^
J
whence, log tan hv= 9.6169771m
iy = 157°30'4r.50
z; = 315 123.00
\ogsJr .... 0.1629939
losr 0.3259878
III. To these methods we add a third which is almc^t equally easy and expe-
ditious, and is much to be preferred to the former if the greatest accuracy should
be required. Thus, ris first determined by means of equation III, and after that,
V by X. Below is our example treated in this manner.
loa;e 9.3897262
cos JE
9.9094637
log sin ^ . . .
log v'(l — ecosE)
9.7663366«
9.9517744
ecos-E'=
9.2991899
0.1991544
log sin h (p
9.8145622^
9.0920395
log a . . . .
log(l — cgobE)
0.4224389
9.9035488
logr 0.3259877
\ogsmh{v — E) . . 8.9066017«
j(y_^)=_4°3r 33-24
v — E r^ — 9 15 6.48
?;c=315 123.02
Formula VIII., or XI., is very convenient for verifying the calculation, par-
ticularly if V and r have been determined by the third method. Thus ;
Sect. 1.] to position in the orbit. 17
log^sin^ . . . 9.8627878^ logsin^V^ . . . 9.8145622?^
log cos 9 .... 9.9865224 logcos^y . . . . 9.9966567
9.8493102^ 9.8112189W
logsiny . . . , 9.8493102» log sin | (y + ^) . . 9.8112189w
15.
Since, as we have seen, the mean anomaly M is completely determined by
means of v and 9, in the same manner as e^ by Jlf and 9, it is evident, that if all
these quantities are regarded as variable together, an equation of condition ought
to exist between their differential variations, the investigation of which will not
be superfluous. By differentiating first, equation YIL, article 8, we obtain
^E ^ dy _
sin^ sinu cosgj'
by differentiating likewise equation XII., it becomes
dif=(l — eco%E)diE — sin ^ cos 9 d 9.
If we eliminate d^from these differential equations we have
dJf=: ^-^ '-^v — (sm^cosg)H ^^ ^jd^j
sm V \ ' ' cos go / '
or by substituting for sin E,\ — e cos E, their values from equations VIII., III.,
dJf:=-^:^de>-"^'-+^>r"dy,
a a cos QD a a cos** g) '
or lastly, if we express both coefficients by means of v and 9 only,
(^j/__ cos^9 ^„j (2-j-ecost))sinycos^qp^
(l-j-ficosw)^ (l-}-ecosv)^ ''
Inversely, if we consider z; as a function of the quantities M, cp, the equation has
this form : —
^^^ggcosg,^^ , (2 + .cosz.)smz;^
rr ' cos qp ^ '
or by introducing E instead o{ v
dv = ^^^dM-{-^^^i2—ecosE—ee)smEd(p.
18 RELATIONS PERTAINING SDIPLY [BoOK I.
16.
The radius vector r is not fully determined by v and 9, or by M and g), but
depends, besides these, upon p or a; its differential, therefore, ■will consist of three
parts. By differentiating equation IT. of article 8, we obtain
dr dp
u. / VI t/ I c 0111 V 1
— =.-i--\-—-. -dy-
v> -n • 1 -4— p. pn^ 7J
1 -|- e cos V 1 -[- e cos v '
By putting here
— = 2 tan cp d CO
pa ' ^
(which follows from the differentiation of equation I.), and expressing, in con-
formity with the preceding article, d y by means of d i!f and d (p, we have, after
making the proper reductions,
dr da , a . . i i,t a ^
— = h- tan wsmvaM cos cp cos vdw,
r a ' r r ' '
dr = - da-l-«tan9)sin^>dJf — a cos 9 cos yd 9.
Finally, these formulas, as well as those which we developed in the preceding
article, rest upon the supposition that v, cp, and M, or rather dv, d 9, and d M,
are expressed in parts of the radius. If, therefore, we choose to express the vari-
ations of the angles v, (p, and M, in seconds, we must either divide those parts of
the formulas which contain dv,d(p,or dM,hy 206264.8, or multiply those which
contain d r, dp, d a, by the same number. Consequently, the formulas of the pre-
ceding article, which in this respect are homogeneous, will require no change.
17.
It will be satisfactory to add a few words concerning the investigation of the
greatest equation of the centre. In the first place, it is evident in itself that the dif-
ference between the eccentric and mean anomaly is a maximum for ^= 90°,
where it becomes = e (expressed in degrees, etc.) ; the radius vector at this point
= a, whence v = 90° -|- y, and thus the whole equation of the centre = 9) -]- e,
Sect. 1.] to position m the orbit. 19
which, nevertheless, is not a maximum here, since the difference between v and
U may still increase beyond cp. This last difference becomes a maximum for
d (y ^) = 0 or for dv = dU, where the eccentricity is clearly to be regarded
as constant. "With this assumption, since in general
dv dU
sin V sin U '
it is evident that we should have sin ^J = sin ^ at that point where the difference
between v and ^ is a maximum ; whence we have by equations VIIL, HI,
r^=acos<p, e cos^= 1 — cos 9, or cos^^-f"*^^ ^ 9-
In like manner cos z; = — tan i cp is found, for which reason it will follow * that
V = 90° -f- arc sin tan ^9, ^ = 90° — arc sin tan | (f ;
hence again
sin^ = v/ (1 — tan^ i (p) = ^^,
SO that the whole equation of the centre at this point becomes
2 arc sin tan i 9 -|- 2 sin ^ 9) \/ cos (f,
the second term being expressed in degrees, etc. At that point, finally, where
the whole equation of the centre is a maximum, we must have dy = dil/^ and
so according to article 15, r=a^ cos (p ; hence we have
1 COS^ op rt 1 \/ COS CD 1 COS (Y) tajQ i QD
cose; = ^, cos-E^== "^ -= .., , , ^ = , , , ,
e ' e e {1 -\- ^ cos cp) l-j-ycosg;'
by which formula U can be determined with the greatest accuracy. U being
found, we shall have, by equations X.,XII.,
equation of the centre = 2 arc sin ^^"/ ^ ^^" 1- e sin U.
\J COS(p
"We do not delay here for an expression of the greatest equation of the centre by
means of a series proceeding according to the powers of the eccentricities, which
several authors have given. As an example, we annex a view of the three
maxima which we have been considering, for Juno, of which the eccentricity,
according to the latest elements, is assumed = 0.2554996.
* It is not necessary to consider those maxima which lie between the aphelion and perihelion,
because they evidently differ in the signs only from those which are situated between the perihelion and
aphelion.
20
RELATIONS PERTAINING SIMPLY
[Book I.
Maximum.
E
E—M
v—E
v—M
E—M
v—E
v — M
90° 0' 0"
82 32 9
86 14 40
14° 38' 20".57
14 30 54 .01
14 36 27 .39
14°48'11".48
14 55 41 .79
14 53 49 .57
29°26'32".05
29 26 35 .80
29 30 16 .96
18.
In the PARABOLA, the eccentric anomaly, the mean anomaly, and the mean
motion, become = 0 ; here therefore these ideas cannot aid in the comparison of
the motion with the time. In the parabola, however, there is no necessity for an
auxiliary angle in integrating rrdv; for we have
4cos^
2 cos^ 4 V
' PP (1 -|-tan^ hv) dtan hv;
and thus,
frr^v = hpp (tan hv -\- k tan^ ^ ^) + Constant.
If the time is supposed to commence with the perihelion passage, the Constant
= 0 ; therefore we have
tani.+ Han^i. = ^-^^^^^+^,
by means of which formula, t may be derived from v, and v from t, when p and
(Jb are known. In the parabolic elements it is usual, instead of p, to make use of
the radius vector at the perihelion, which is h p, and to neglect entirely the mass
(I. It will scarcely ever be possible to determine the mass of a body, the orbit of
which is computed as a parabola ; and indeed all comets appear, according to the
best and most recent observations, to have so little density and mass, that the
latter can be considered insensible and be safely neglected.
19.
The solution of the problem, from the true anomaly to find the time, and, in
a still greater degree, the solution of the inverse problem, can be greatly abbrevi-
ated by means of an auxiliary table, such as is found in many astronomical works.
Sect. 1.] to position in the orbit. 21
But the Barkerian is by far the most convenient, and is also annexed to the
admirable work of the celebrated Olbeks, [Ahlmndlung uber die leichteste und
leqiiemste Methode die Bahn eines Cometen su lerechnen : Weimar, 1797.) It contains,
under the title of the mean motion, the value of the expression 75 tan ^y-]- 25
tan^ ^ V, for all true anomalies for every five minutes from 0 to 180°. If
therefore the time corresponding to the true anomaly v is required, it will be
necessary to divide the mean motion, taken from the table with the arguments,
150 h •
by — ^, which quantity is called the mean daily motion; if on the contrary the
true anomaly is to be computed from the time, the latter expressed in days will
be multiplied by — ^, in order to get the mean motion, with which the correspond-
ing anomaly may be taken from the table. It is further evident that the same
mean motion and time taken negatively correspond tp the negative value of the v ;
the same table therefore answers equally for negative and positive anomalies. If
in the place of />,we prefer to use the perihelion distance \'p-=zq, the mean daily
^ v/2812 5
motion is expressed by ^ ^ ' , in which the constant factor ^y/ 2812,5 =^
0.912279061, and its logarithm is 9.9601277069. The anomaly v being found,
the radius vector will be determined by means of the formula already given.
20.
By the differentiation of the equation
tan \v-\-l tan^ ^ y = 2 t'kp~^,
if all the quantities v, t,p, are regarded as variable, we have
22 RELATIONS PERTAIMNG SBIPLY [BoOK I.
If the variations of the anomaly v are wanted in seconds, both parts also of
^v must be expressed in this manner, that is, it is necessary to take for k the value
3548".188 given in article 6. If, moreover, ^p=q is introduced instead of p, the
formula wiU have the following form :
rr rrsjzq ^'
in which are to be used the constant logarithms
log ks/2 = 3.7005215724, log^/c^i = 3.8766128315.
Moreover the differentiation of the equation
furnishes
— = —-I- tan ivdiV,
or by expressing dv by means of d^ and dp,
Ar /I Zht\a.n\v\ ■, j^lc\jpian\v ■, ,
r \p %rr^p /-'"'" ^.^
By substituting for i its value in v, the coefficient of dp is changed into
1 _ 3^Un^ _ ^jan^ _ 1 / ^^^, .^^
but the coefl&cient of d^ becomes — t—* From this there results
or if we introduce q for p
z 1 ;i , ^ sin w , ,
d r = i cos z; d j» -[" "w — " ^>
d, , ^ sin « J ,
r ^ COS y d g* -|- ,^ dt.
The constant logarithm to be used here is log^ y' I = 8.0850664436.
21.
In the HYPERBOLA, 9 and U would become imaginary quantities, to avoid
which, other auxiliary quantities must be introduced in the place of them. We
have already designated by ijJ the angle of which the cosine =-, and we have
found the radius vector
Sect. 1.] TO position in the orbit. 23
2ecos^(v — ilj) cos^(v-{-xp)'
For v=0, the factors cos i (v — xp), and cos ^ (v -\-xp), in the denominator of this
fraction become equal, the second vanishes for the greatest positive value of v,
and the first for the greatest negative value. Putting, therefore,
cos ^(v — -w)
cos I- (v + 'V^) '
we shall have m = 1 in perihelion ; it will increase to infinity as v approaches its
limit 180° — If; on the other hand it will decrease indefinitely as v is supposed
to return to its other limit — (180° — if) ; so that reciprocal values of ««, or, what
amounts to the same thing, values whose logarithms are complementary, corre-
spond to opposite values of v.
This quotient u is very conveniently used in the hyperbola as an auxiliary
quantity ; the angle, the tangent of which is
can be made to render the same service with almost equal elegance ; and in order
to preserve the analogy with the ellipse, we will denote this angle by ^ F. In
this way the following relations between the quantities v, r, ti, F are easily brought
together, in which we put a = — ^, so that h becomes a positive quantity.
I. J=j»cotan^if
n. r = ^ ^= pcostft
l-j-ecosv 2cos^(v — \pi)(ios^{v-\--\p)
m. tan^i?'=tan^yt/^ = tan^ytan|w = ^^
Y 1 = i Tm J- 1^ 1 -|- COS 1/> COS t> e -j- COS ^
cos^ ^ ~^ u' 2cos^(y — 1/;) cos ^ («;-[- If) l + ecosv*
By subtracting 1 from both sides of equation V. we get,
24 KELATIONS PERTAINING SIMPLY [BoOK I.
In the same manner, by adding 1 to both sides, it becomes
VII. cosiW = oosiJ'y/^j,p^^^=cosiJy/(fj=i)i
By dividing VI. by VII. we should reproduce III. : the multiplication produces
VIII. rsmv=p cotan if tan F=i tan f tan F
= hp cotan i^ [u )=z ^ h tan t// [u ) .
From the combination of the equations II. V. are easily derived
X. r co^v =zh (e =,)-=^lh(2e — u ),
22.
By the differentiation of the formula IV. (regarding if as a constant quantity)
we get
— = i (tan h (y-j- V) — ^^^ ^ (^ — tf)jd2>:=^ ^°^de;;
hence,
u tan 1/) '
or by substituting for r the value taken from X.
rrde;=: JJtam/^ \\ e{\ -\ ) Mm.
Afterwards by integrating in such a manner that the integral may vanish at the
perihelion, it becomes
/rrdy^^Hani// {\e{u ) — \ogv)=^ht\l psjiX-^- \i)=:.'kt\2X^v;i sjlsj {\-\- \i).
The logarithm here is the hyperbolic j if we wish to use the logarithm from
Brigg's system, or in general from the system of which the modulus = \, and
Sect. 1.] to position in the orbit. 25
the mass fi (which we can assume to be indeterminable for a body moving in an
hyperbola) is neglected, the equation assumes the following form : —
XL ne'^^^^ — logu^^-^,
or by introducing F,
leianF—\ogtsin{4:6°-{-iF) = ~.
Supposing Brigg's logarithms to be used, we have
log I = 9.6377843113,, log 11-=^ 7.8733657527 ;
but a little greater precision can be attained by the immediate application of the
hyperbolic logarithms. The hyperbolic logarithms of the tangents are found in
several collections of tables, in those, for example, which Schulze edited, and still
more extensively in the Magnus Canon Triangulor. LogaritJimiciis of Benjamin Uksin,
Cologne, 1624, in which they proceed, by tens of seconds.
Finally, formula XI. shows that opposite values of t correspond to reciprocal
values of u, or opposite values of F and v, on which account equal parts of the
hyperbola, at equal distances from the perihelion on both sides, are described in
equal times.
23.
If we should wish to make use of the auxiliary quantity u for finding the
time from the true anomaly, its value is most conveniently determined by means
of equation IV. ; afterwards, formula II. gives directly, without a new calculation,
p by means of r, or r by means of p. Having found u, formula XL will give the
quantity -^, which is analogous to the mean anomaly in the ellipse and will be
denoted by N, from which will follow the elapsed time after the perihelion transit.
Since the first term of JSf, that is ^ ^u" ^^^^^ ^J means of formula Vm. be
made = /f^"", the double computation of this quantity will answer for testing
its accuracy, or, if preferred, N can be expressed without u, as follows : —
XII. iV= Itmypsmv ^^^cos\{v — ^p)
2 cos i (f -f- If)) cos ^{v — Mj) ^ cos i (v + -U") '
4
26
RELATIONS PERTAINING SBIPLY
[Book I.
Example. — Let e= 1.2618820, or iy = 37° 35' 0'', v = 18° 51' 0", log r =
0.0333585. Then the computation for ii,p, h, N, t, is as follows: —
log cos J(y — 1//) . . 9.9941706)
logcos-i(y + i/;) . . 9.9450577)
logr 0.0333585
log 2 6 0.4020488
logj? . . ,
log cotan^ If
0.3746356
0.2274244
loo; h
0.6020600
9.4312985
log sin y 9.5093258
logX 9.6377843
Compl. log sin T/7 . . 0.2147309 '
8.7931395
First term of iV^=r . 0.0621069
logw= 0.0491129
N= 0.0129940
log^^ 7.8733658)
flog 5 0.9030900)
hence,
)gu . .
. 0.0491129
^^ =
1.1197289
uu^=
1.2537928
The other calculation.
log(?a( — 1) . . . 9.4044793
Compl. logM . . . 9.9508871
log ^ 9.6377843
logle 9.7999888
8.7931395
logiV 8.1137429
Difference .... 6.9702758
logt 1.1434671
t = 13.91448
24
If it has been decided to carry out the calculation with hyperbolic logarithms,
it is best to employ the auxiliary quantity F, which will be determined by equa-
tion III., and thence JV by XI. ; the semi-parameter will be computed from the
radius vector, or inversely the latter from the former by formula YIII. ; the
second part of JV can, if desired, be obtained in two ways, namely, by means of the
formula hyp. log tan (45° -\- ^ F), and by this, hyp. log cos h {v — t//) — hyp. log
cos I [v -\- T/'). Moreover it is apparent that here where ^ = 1 the quantity N
Sect. 1.]
TO POSITION IN THE ORBIT.
27
will come out greater in the ratio 1 : X, than if Brigg's logarithms were used.
Our example treated according to this method is as follows : —
lug uaii 2 1// ....
log tan kv
9.2201009
log tan ^^ . . . .
8.7519188
loa; e
0.1010188
log tan ^
9.0543366
e tan F —
9.1553554
0.14300638
hyp. log tan (45° -]- i J') =
= 0.11308666
N—
0.02991972
loffJc
8.2355814 )
flogJ
0.9030900 i
^ J^=3°13'58'a2
C. hyp. log cos h{v — 1/')
C. hyp. log cos i [v -\-i^):
0.01342266
0.12650930
Difference
0.11308664
logiV^ 8.4759575
Difference 7.3324914
\ogt 1.1434661
t= 13.91445
25.
For the solution of the inverse problem, that of determining the true anomaly
and the radius vector from the time, the auxiliary quantity u or F must be first
derived from iV=: IkV^t by means of equation XI. The solution of this tran-
scendental equation will be performed by trial, and can be shortened by devices
analogous to those we have described in article 11. But we suffer these to pass
without further explanation ; for it does not seem worth while to elaborate as
carefully the precepts for the hyperbolic motion, very rarely perhaps to be exhib-
ited in celestial space, as for the elliptic motion, and besides, all cases that can
possibly occur may be solved by another method to be given below. After-
wards F or u will be found, thence v by formula III., and subsequently r will be
determined either by II. or VIII. ; v and r are still more conveniently obtained
by means of formulas VI. and VII. ; some one of the remaining formulas can be
called into use at pleasure, for verifying the calculation.
28
KELATIONS PERTAINING SIMPLY
[Book I.
26.
Example. — Eetaining for e and I the same values as in the preceding example,
let t = 65.41236 : v and r are required. Using Briggs's logarithms we have
log?! 1.8166598
log^^J-l .... 6.9702758
log i\^ 8.7859356, whence N= 0.06108514. Prom this it is
seen that the equation iV=^etan^ — log tan (45° -|- ^ i^) is satisfied by
F=:i 25°24'27".66, whence we have, by formula III,
log tan ii^ . . . . 9.3530120
log tan ^Y . . . . 9.5318179
log tan J y .... 9.8211941, and thus ^ y = 33° 31' 29".89, and v —
67° 2' 59';78. Hence, there follows,
C.logco3i(. + ,) . 0.21374T6| ^^^_^^ ^^^^^^^^
C.logcos|(y — -li^) . 0.0145197)
° log tan (45° 4-^^) . . . 0.1992280
log f^ 9.9725868 ^ \ -r ^ )
loff /- . . . . . . 0.2008541.
27.
If equation IV. is differentiated, considering m, v, if, as variable at the same
time, there results,
d M sin 1/; d u -]- sin v^yp r tan tl;
dy +
di//.
u 2 COS -^ (v — 1/;) cos -g- (w -}- If ) p '" ' j9 cos \p
By differentiating in like manner equation XL, the relation between the
differential variations of the quantities u, xp, N, becomes,
= (|,(l_|_i.)_l)dw.- (" — 1)^-^
\ ^ ' uu' uf
or
diV
diV"
2 u cos^ 1/;
di//,
-v~ = 7- d i( + -. d w .
/. OU ^ b cos I/; '
Sect. 1.] to position in the okbit. 29
Hence, by eliminating d u by means of the preceding equation we obtain
-T- =n dv-\- (14- -)t dw,
dbbt&nilJ T TiT /5 I 5 \ sin V tan 1/; ^
v = —, — ^di\^ — {- + -) T^dw
Xrr \r ^ p / cost/' '
= —, — ^diV^ — (14- ^)^— d^//.
Xrr \ ' r/sini/j '
28.
By differentiating equation X., all the quantities r, h, e, u, being regarded as
variables, by substituting
^ sin t/» T
ae = — ^ aw,
cos^xp ' '
and eliminating du with the help of the equation between dJY, du, dijj, given in
the preceding article, there results.
dr = Td^
bbe(uu — 1)
b ' 2lur
^^+2^{(^' + ^)«^^^-(^^-.^)'^^^|^^-
The coefficient of diV is transformed, by means of equation YIII., into ,-^^^ : but
the coefficient of d Tf , by substituting from equation IV.,
u (sin y — sine;) ^ sin (i// — v), - (sin 1/; -j- sin v) = sin {ip -{- v),
is changed into
h sin xp cos v pcosv ^
cos^i/; siaxp '
so that we have
-, r J , , 5 sin w -, ,7- , » cos v ,
0 ' I sin ip ' sm V '
So far, moreover, as JY is considered a function of b and t, we have
dJY=^df-^^db,
which value being substituted, we shall have d r, and also d y in the preceding
article, expressed by means of d ^f, d ^, d if. Finally, we have here to repeat our
30 RELATIONS PERTAINING SMPLY [BoOK 1.
previous injunction, that, if the variations of the angles v and v^ are conceived to
be expressed, not in parts of the radius, but in seconds, either all the tenns con-
taining d y, d T//, must be divided bj 206264.8, or all the remaining terms must be
multiplied by this number.
29.
Since the auxiliary quantities y, E, M, employed in the ellipse obtain
imaginary values in the hyperbola, it will not be out of place to investigate their
connection with the real quantities of which we have made use: we add therefore
the principal relations, in which we denote by i the imaginary quantity y^ — 1.
1
sin (p = e =
•* cos \p
tan (45° — I 9) = ^^^ = 2y J^ = /tan i t/;
tan 9) = 1 cotan (45° — Icp) — | tan (45° — 29)= r^
cos Cjp = i tan i\i
9 = 90° -f i log (sin 9 + i cos 9) r= 90° — i log tan (45° + i i/^)
tan ^ ^ = i taniF = li^izil)
- — =-=r ^ cotan i U -I- i tan i U=^ — i cotan F,
sin jQ '
or
or
n -J -n i(uu 1)
2m
jTil Til l+Qni 7?7 —
sinJ''
2m
cotan F=i cotan i F — ^ tan ^ F = —
, rr . . T7 i(uu — 1)
tan^=«sm^=^ — r-~-
uu-\-l
■n 1 MM + 1
cos E = ^ := „ '
COS ^ 2 M
iF= log (cos ^4" *'si^ ^) =
or
F = i log u = i log (45° -\- h F)
° 2m a
The logarithms in these formulas are hyperbolic.
Sect. 1.] to position in the okbit. 31
30.
Since none of the numbers which we take out from logarithmic and trigo-
nometrical tables admit of absolute precision, but are all to a certain extent
approximate only, the results of all calculations performed by the aid of these
numbers can only be approximately true. In most cases, indeed, the common
tables, which are exact to the seventh place of decimals, that is, never deviate
from the truth either in excess or defect beyond half of an unit in the seventh
figure, furnish more than the requisite accuracy, so that the unavoidable errors
are evidently of no consequence : nevertheless it may happen, that in special
cases the effect of the errors of the tables is so augmented that we may be
obliged to reject a method, otherwise the best, and substitute another in its place.
Cases of this kind can occur in those computations which we have just explained;
on which account, it will not be foreign to our purpose to introduce here some
inquiries concerning the degree of precision allowed in these computations by
the common tables. Although this is not the place for a thorough examination
of this subject, which is of the greatest importance to the practical computer, yet
we will conduct the investigation sufficiently far for our own object, from which
point it may be further perfected and extended to other operations by any one
requiring it.
31.
Any logarithm, sine, tangent, etc. whatever, (or, in general, any irrational
quantity whatever taken from the tables,) is liable to an error which may amount
to a half unit in the last figure : we will designate this limit of error by w, which
therefore is in the common tables = 0.00000005. If now, the logarithm, etc.,
cannot be taken directly from the tables, but must be obtained by means of inter-
polation, this error may be slightly increased from two causes. In the first place, it is
usual to take for the proportional part, when (regarding the last figure as unity) it
is not an integer, the next greatest or least integer ; and in this way, it is easily
perceived, this error may be increased to just within twice its actual amount. But
32 KELATIONS PERTALNING SDIPLY [BoOK I,
we shall pay no attention to this augmentation of the error, since there is no
objection to our affixing one more than another decimal figure to the propor-
tional part, and it is verj evident that, if the proportional part is exact, the inter-
polated logarithm is not liable to a greater error than the logarithms given
directly in the tables, so far indeed as we are authorized to consider the changes
in the latter as uniform. Thence arises another increase of the error, that this
last assumption is not rigorously true ; but this also we pretermit, because the
effect of the second and higher differences (especially where the superior tables
computed by Taylor are used for trigonometrical functions) is evidently of no
importance, and may readily be taken into account, if it should happen to turn
out a little too great. In all cases, therefore, we will put the maximum unavoid-
able error of the tables = w, assuming that the argument (that is, the number the
logarithm of which, or the angle the sine etc. of which, is sought) is given with
strict accuracy. But if the argument itself is only approximately known, and
the variation w' of the logarithm, etc. (which may be defined by the method of
differentials) is supposed to correspond to the greatest error to which it is liable,
then the maximum error of the logarithm, computed by means of the tables, can
amount to w -|- (a' .
Inversely, if the argument corresponding to a given logarithm is computed
by the help of the tables, the greatest error is equal to that change in the argu-
ment which corresponds to the variation w in the logarithm, if the latter is cor-
rectly given, or to that which corresponds to the variation w -f- w' in the loga-
rithm, if the logarithm can be erroneous to the extent of w'. It will hardly be
necessary to remark that w and 0/ must be affected by the same sign.
If several quantities, correct within certain limits only, are added together,
the greatest error of the sum will be equal to the sum of the greatest individual
errors affected by the same sign ; wherefore, in the subtraction also of quantities
approximately correct, the greatest error of the difference will be equal to the
sum of the greatest individual errors. In the multiplication or division of a
quantity not strictly correct, the maximum error is increased or diminished in the
same ratio as the quantity itself
Sect. 1.] to position in the orbit. 33
32.
Let us proceed now to the application of these principles to the most useful
of the operations above explained.
I. If cp and ^ are supposed to be exactly given in using the formula VII.,
article 8, for computing the true anomaly from the eccentric anomaly in the
elliptic motion, then in log tan (45° — i (p) and log tan ^^, the error w maybe
committed, and thus in the difference = log tan i v, the error 2 w ; therefore the
greatest error in the determination of the angle i v will be
ZmA\v 3 CO sin w
d log tan \ V 2 X '
X denoting the modulus of the logarithms used in this calculation. The error,
therefore, to which the true anomaly v is liable, expressed in seconds, becomes
^^^' 206265 = 0".0712 sin v,
if Bri^g's logarithms to seven places of decimals are employed, so that we may
be assured of the value of y within 0".07 ; if smaller tables to five places only, are
used, the error may amount to 7'M2.
II. If e cos E is computed by means of logarithms, an error may be committed
to the extent of
3 CO e cos ^
I '
therefore the quantity
1 — ecosK or - ,
will be liable to the same error. In computing, accordingly, the logarithm of this
quantity, the error may amount to (1 -{-(?) w, denoting by d the quantity
3 e cos J?
1 — e cos ^
taken positively : the possible error in log r goes up to the same limit, log a being
assumed to be correctly given. If the eccentricity is small, the quantity d is
always confined within narrow limits; but when e differs but little from 1,
1 — ecosE remains very small as long as E is small ; consequently, 8 may
5
34 RELATIONS PERTAINING SBIPLY [BoOK I.
increase to an amount not to be neglected : for this reason formula III., article 8,
is less suitable in this case. The quantity <^ may be expressed thus also,
3 (a — r) 3 e (cos v-\-e)
r 1 — ee '
which formula shows still more clearly when the error (1 -|-^) w may be neglected.
III. In the use of formula X., article 8, for the computation of the true from
the mean anomaly, the logu- is liable to the error {^ -{- ^d)o), and so the log
sin ^(psmU u - to that of ( | -|- i^ <5') co ; hence the greatest possible error in the
determination of the angles v — ^ or y is
or expressed in seconds, if seven places of decimals are employed,
(O'aee + 0".024 d) tan^v — U).
When the eccentricity is not great, d and tan ^ {v — U) will be small quantities,
on account of which, this method admits of greater accuracy than that which
we have considered in I. : the latter, on the other hand, will be preferable
when the eccentricity is very great and approaches nearly to unity, where d and
tan i {v — U) may acquire very considerable values. It will always be easy to
decide, by means of our formulas, which of the two methods is to be preferred.
IV. In the determination of the mean anomaly from the eccentric by means
of formula XII., article 8. the error of the quantity e sin U, computed by the help
of logarithms, and therefore of the anomaly itself, M, may amount to
3 cue sin ^
■ I '
which limit of error is to be multiplied by 206265" if wanted expressed in
seconds. Hence it is readily inferred, that in the inverse problem where U is to
be determined from 31 by trial, U may be erroneous by the quantity
^J!llp^, ^^. 206265^^::=^'" 7^'^"--. 206265",
even if the equation E — esin^= 3/ should be satisfied with all the accuracy
which the tables admit.
Sect. 1.]
TO POSITION IN THE ORBIT.
35
The true anomaly therefore computed from the mean may be incorrect in
two ways, if we consider the mean as given accurately; first, on account of the
error committed in the computation of v from E, which, as we have seen, is of
slight importance ; second, because the value of the eccentric anomaly itself may
be erroneous. The effect of the latter cause will be expressed by the product of
the error committed in E into -r^, which product becomes
^"7°^. i^. 206265" = ^-^^^^. 206265'' = ('-^^^^^til^^^^")0''.0712,
if seven places of decimals are used. This error, always small for small values of
e, may become very large when e differs but little from unity, as is shown by the
following table, which exhibits the maximum value of the preceding expression
for certain values of e.
e
maximum error.
e
maximum error.
e
maximum eiTor.
0.90
0.91
0.92
0.93
0".42
0.48
0.54
0.62
0.94
0.95
0.96
0.97
0".73
0.89
1 .12
1 .50
0.98
0.99
0.999
2".28
4.59
46.23
V. In the hyperbolic motion, if v is determined by means of formula III,
article 21, from F and if accurately known, the error may amount to
S_^pv^ 206265'^;
but if it is computed by means of the formula
tan i V ■■
(u — 1) tani-i/;
M+1 '
u and tf being known precisely, the limit of the error will be one third greater,
that is.
4 o) sin V
for seven places.
VI. If the quantity
. 206265" = 0':09 sin y
— — iV"
is computed by means of formula XI., article 22, with the aid of Briggs's loga-
36 RELATIONS PERTAINING SIMPLY [BoOK I.
rithms, assuming e and u or e and F to be known exactly, the first part will be
liable to the error
5 (mm — l)e(u
2^^ »
if it has been computed in the form
or to the error
if computed in the form
Ae(M — 1)(m + 1)^
2« ^
3 (mm-|-1) ecu
n^M-^;
or to the error Sew tan F if computed in the form X e tan jP, provided we neglect
the error committed in log X or log i h In the first case the error can be
expressed also by 5 e w tan F, in the second by 7^, whence it is apparent that
the error is the least of all in the third case, but will be greater in the first or
second, according as ^< or - ^^ 2 or <^ 2, or according as +-^^ 36° 52' or <[ 36° 52'.
But, in any case, the second part of N will be liable to the error w,
VII. On the other hand, it is evident that if m or J' is derived from iV^ by
trial, u would be liable to the error
(w + 5 e to tan F) -r^,
or to
/ , Zeoi ^ Au
according as the first term in the value of N is used separated into factors, or into
terms ; F, however, is liable to the error
(tu + 3 e w tani^) j-^ .
The upper signs serve after perihelion, the lower before perihelion. Now if
■^ is substituted here for -r^ or for -tjt, the effect of this error appears in
the determination of v, which therefore will be
Sect. 1.]
TO POSITION IN THE ORBIT.
37
hhtamp (l±Se tan F) w hh tamp (1 -f- 3 e secF) co
Irr Xrr '
if the auxiliary quantity u has been employed ; on the other hand, if F has been
used, this effect becomes,
J 5 tan 1/^ (1 + 3 e tan jP) CO a ( (1 -|- e cos vy ,
Irr X \ tan**!/; —
3 e sin tJ (1 -|- e cos v)
tan^tl;
If the error is to be expressed in seconds, it is necessary to apply the factor
206265". It is evident that this error can only be considerable when i/^ is a small
angle, or e a little greater than 1. The following are the greatest values of this
third expression, for certain values of e, if seven places of decimals are employed:
e
maximum error.
1.3
0".34
1.2
0.54
1.1
1.31
1.05
3 .03
1.01
34.41
1.001
1064 .65
To this error arising from the erroneous value of F or u it is necessary to
apply the error determined in Y. in" order to have the total uncertainty of v.
VIII. If the equation XI., article 22, is solved by the use of hyperbolic loga-
rithms, F being employed as an auxiliary quantity, the effect of the possible
error in this operation in the determination of v, is found by similar reasoning
to be,
(1 -)- e cos w)^ w' I 3 e sin w (1 -j- e cos zj) ft)
tan^ yp — I tan^ ■^j '
where by oi' we denote the greatest uncertainty in the tables of hyperbolic logar
rithms. The second part of this expression is identical with the second part of
the expression given in VIL; but the first part in the latter is less than the first
in the former, in the ratio I to' : w, that is, in the ratio 1 : 23, if it be admissible
to assume that the table of Ursin is everywhere exact to eight figures, or
oi' == 0.000000005.
38 KELATIONS PERTAINI2vG SBIPLY [BoOK I.
33.
The methods above treated, both for the determination of the true anomaly
from the time and for the determination of the time from the true anomaly,* do
not admit of all the precision that might be required in those conic sections of
which the eccentricity differs but little from unity, that is, in ellipses and hyper-
bolas which approach very near to the parabola; indeed, unavoidable errors,
increasing as the orbit tends to resemble the parabola, may at length exceed all
limits. Larger tables, constructed to more than seven figures would undoubtedly
diminish this uncertainty, but they would not remove it, nor would they prevent
its surpassing all limits as soon as the orbit approached too near the parabola.
Moreover, the methods given above become in this case very troublesome, since a
part of them require the use of indirect trials frequently repeated, of which
the tediousness is even greater if we work with the larger tables. It certainly,
therefore, will not be superfluous, to furnish a peculiar method by means of
which the uncertainty in this case may be avoided, and sufficient precision may
be obtained with the help of the common tables.
34.
The common method, by which it is usual to remedy these inconveniences,
rests upon the following principles. In the ellipse or hyperbola of which e is the
eccentricity, 'p the semi-parameter, and therefore the perihelion distance
let the true anomaly v correspond to the time t after the perihelion; in the
parabola of which the semi-parameter = 2 ^, or the perihelion distance = q, let
the true anomaly w correspond to the same time, supposing in each case the
mass \x, to be either neglected or equal. It is evident that we then have
* Since the time contains the factor a^ or S^, the greater the values of a = , ^ - , or 5= ^ ,
" 1 — ee e^ — 1
the more the error in illf or i^Twill be increased.
Sect. 1.] to position in the orbit. 39
J (l+ecosi;)2- j (1+cosm;)2 —\F'M^1^
the integrals commencing from ^ = 0 and tv ^ 0, or
J (l+ecosw)V2 J (l+cos«;)2
Denoting y-^ by a, tan ^vhy&, the former integral is found to be
y/(l_^a). (^^_^|^3(i_2«)_|^5(2a_3cca) + |^^(3«a — 4«3) — etc.),
the latter, tan h w -\- Han^ ^ iv. From this equation it is easy to determine to
by a and t', and also vhy a and ^^ by means of infinite series : instead of a may
be introduced, if preferred,
-1 2 « jt,
1 — ez=i--T— ^=0.
\-\-a
Since evidently for « = 0, or (^ = 0, we have v^=-w, these series will have the
following form : —
w = v-\-dv' -^ddv" ^ dH'" + etc.
where v', v", v'", etc. will be functions of v, and tv', tv\ iv" , functions of iv. When
d is a very small quantity, these series converge rapidly, and few terms suffice for
the determination of w from v, or of v from iv. i is derived from iv, or lo from t,
by the method we have explained above for the parabolic motion.
35.
Our Bessel has developed the analytical expressions of the three first coeffi-
cients of the second series iv', w" , iv'", and at the same time has added a table con-
structed with a single argument lo for the numerical values of the two first lo'
and iv", ( Von Zach Moimtliclie Correspondenz, vol. XIL, p. 197). A table for the
first coefficient w', computed by Simpson, was already in existence, and was
annexed to the work of the illustrious Olbers above commended. By the use
of this method, with the help of Bessel's table, it is possible in most cases to
determine the true anomaly from the time with sufficient precision; what remains
to be desired is reduced to nearly the following particulars : —
40 EELATIONS TERTAINIXG SQIPLY [BoOK I.
I. In the inverse problem, the determination of the time, that is, from the
true anomaly, it is requisite to have recourse to a somewhat indirect method, and
to derive iv from v by trial. In order to meet this inconvenience, the first series
should be treated in the same manner as the second : and since it may be readily
perceived that — v' is the same function of v as to' of tv, so that the table for iv
might answer for v' the sign only being changed, nothing more is required than
a table for v", by which either problem may be solved with equal precision.
Sometimes, undoubtedly, cases may occur, where the eccentricity differs but
little from unity, such that the general methods above explained may not appear
to afford sufficient precision, not enough at least, to allow the effect of the third
and higher powers of d in the peculiar method just sketched out, to be safely
neglected. Cases of this kind are possible in the hyperbolic motion especially, in
which, whether the former methods are chosen or the latter one, an error of
several seconds is inevitable, if the common tables, constructed to seven places of
figures only, are employed. Although, in truth, such cases rarely occur in prac-
tice, something might appear to be wanting if it were not possible in all cases to
determine the true anomaly within 0".l, or at least 0".2, without consulting the
larger tables, which would require a reference to books of the rarer sort. We
hope, therefore, that it will not seem wholly superfluous to proceed to the exposi-
tion of a peculiar method, which we have long had in use, and which will also
commend itself on this account, that it is not limited to eccentricities differing but
little from unity, but in this respect admits of general application.
36.
Before we proceed to explain this method, it will be proper to observe that
the uncertainty of the general methods given above, in orbits approaching the
form of the parabola, ceases of itself, when E or F increase to considerable mag-
nitude, which indeed can take place only in large distances from the sun. To
show which, we give to
So^eosin.^ 206265'',
the greatest possible error in the ellipse, which we find in article 32, IV., the
following form,
Sect. 1.]
TO POSITION IN THE ORBIT.
41
<)e\l (1 — ee). s\nE
l{\—ecosEY •
206265''
from which is evident of itself that the error is always circumscribed within
narrow limits when E acquires considerable value, or when cos E recedes further
from unity, however great the eccentricity may be. This will appear still more
distinctly from the following table, in which we have computed the greatest
numerical value of that formula for certain given values of E, for seven decimal
places.
^=10°
maximum error
= rM
20
0.76
30
■ 0 .34
40
0 .19
50
0 .12
60
0 .08
The same thing takes place in the hyperbola, as is immediately apparent, if the
expression obtained in article 32, VII., is put into this form,
1)
CO cos-F (cos^-|- 3 e sin F) \J (e
' l(e — cosFy
206265".
The following table exhibits the greatest values of this expression for certain
given values of E.
F
«
maximum error.
10°
1.192
0.839
8".66
20
1.428
0.700
1 .38
30
1.732
0.577
0.47
40
2.144
0.466
0.22
50
2.747
0.364
0.11
60
3.732
0.268
0.06
70
5.671
0.176
0.02
When, therefore, E or E exceeds 40° or 50° (which nevertheless does not easily
occur in orbits differing but little from the parabola, because heavenly bodies
moving in such orbits at such great distances from the sun are for the most part
withdrawn from our sight) there will be no reason for forsaking the general
method. For the rest, in such a case even the series which we treated in article
42 EELATIONS PERTAINING SQITLY [BoOK I.
34 might converge too slowly ; and therefore it is by no means to be regarded
as a defect of the method about to be explained, that it is specially adapted
to those cases in which E or F has not yet increased beyond moderate values.
37.
Let us resume in the elliptic motion the equation between the eccentric
anomaly and the time,
E — e sm ^ = - V V -rr/
where we suppose E to be expressed in parts of the radius. Henceforth, we
shall leave out the factor Y^(l-|-i"'); if a case should occur where it is worth
while to take it into account, the symbol t would not express the time itself after
perihelion, but this time multiplied by y'(l -j-j"-)- We designate in future by q the
perihelion distance, and in the place of j&and sin ^, we introduce the quantities
^— sin^, and E— J^ [E— sin E) = J^ ^-j- J^ sin ^:
the careful reader will readily perceive from what follows, our reason for selecting
particularly these expressions. In this way our equation assumes the following
form : —
(l-.)(^V^+TVsin^) + (xV + TV^)(^-sm^) = ^^C-^)^
As long as E is regarded as a quantity of the first order,
T'o^+TVsin^=^— eV^' + TiVo^' — etc.
will be a quantity of the first order, while
^— sin^=i^« — ^io^'+5oVo^' — etc.,
will be a quantity of the third order. Putting, therefore,
4^=^^— 3V^'-WTo^'-etc.
will be a quantity of the second order, and
-^=1 + 2/0(7^'- etc.
will differ from unity by a quantity of the fourth order. But hence our equation
becomes
Sect. 1.]
TO POSITION IN THE ORBIT.
^(2(l-.)^^-+^V(l-f9.)/) = ^"^(^)^
43
[1]
By means of the common trigonometrical tables, j\ E -f- iV ^i^ ^ "^^7 be com-
puted with sufficient accuracy, but not E — sin ^ when ^ is a small angle; in this
way therefore it would not be possible to determine correctly enough the quan-
tities A and B. A remedy for this difficulty would be furnished by an appro-
priate table, from which we could take out with the argument E, either B or the
logarithm of B ; the means necessary to the construction of such a table will
readily present themselves to any one even moderately versed in analysis. By
the aid of the equation
9^-f sin^
20 5
U,
^ A can be determined, and hence t by formula [1] with all desirable precision.
The following is a specimen of such a table, which will show the slow increase
of log B ; it would be superfluous to take the trouble to extend this table, for
further on we are about to describe tables of a much more convenient form.
E
log 5
E
log 5
E
logB
0°
0.0000000
25°
0.0000168
50°
0.0002675
5
00
30
0349
55
3910
10
04
35
0645
60
5526
15
22
40
1099
20
69
45
1758
38.
It will not be useless to illustrate by an example what has been given in the
preceding article. Let the proposed true anomaly =z 100°, the eccentricity
= 0.96764567, log q = 9.7656500. The following is the calculation for E, B,
A, and t : —
log tan ^y 0.0761865
logv/^ 9.1079927
^ ~r ^
log tan J^ 9.1841792, whence I ^= 8°4ri9''.32, and ^ =
44 RELATIONS PERTAINING SMPLY [BoOK I.
17° 22' 38".64. To this value of E corresponds log B = 0.0000040 ; next is found
in parts of the radius,^ = 0.3032928, sin E= 0.2986643, whence 2^ ^+ oV sin^
= 0.1514150, the logarithm of which = 9.1801689, and so log 4* = 9.1801649.
Thence is derived, by means of formula [1] of the preceding article,
'»g.T(?i- • • 2.4589614 ^o^-^^^J. . . 3.7601038
1
logA^ 9.1801649 log^^ 7.5404947
log 43.56386= . . 1.6391263 log 19.98014= 1.3005985.
19.98014
63.54400 = !{.
If the same example is treated according to the common method, e sin E in
seconds is found = 59610".79 = 16°33'30".79, whence the mean anomaly =
49'7''.85 = 2947'^85. And hence from
log /?: (^-^')^ = 1.6664302
is derived t = 63.54410. The difference, which is here only yo^o'o' P^^'^ ^^ ^ ^^7?
might, by the errors concurring, easily come out three or four times greater.
It is further evident, that with the help of such a table for log B even the inverse
problem can be solved with all accuracy, E being determined by repeated trials,
so that the value of t calculated from it may agree with the proposed value.
But this operation would be very troublesome : on account of which, we will now
show how an auxiliary table may be much more conveniently arranged, indefinite
trials be altogether avoided, and the whole calculation reduced to a numerical
operation in the highest degree neat and expeditious, which seems to leave
nothing to be desired.
39.
It is obvious that almost one half the labor which those trials would require,
could be saved, if there were a table so arranged that log B could be immedi-
ately taken out with the argument A. Three operations would then remain ;
the first indirect, namely, the determination of A so as to satisfy the equation
Sect. 1.] to position in the oebit. 45
[1], article 37 ; the second, the determination of U from A and B, which may be
done directly, either by means of the equation
or by this,
sin^=2^(4* — 1^*);
the third, the determination of v from B by means of equation YII., article 8.
The first operation, we will bring to an easy calculation free from vague trials ;
the second and third, we will really abridge into one, by inserting a new quantity
C in our table by which means we shall have no need of B, and at the same
time we shall obtain an elegant and convenient formula for the radius vector.
Each of these subjects we will follow out in its proper order.
First, we will change the form of equation [1] so that the Barkerian table
may be used in the solution of it. For this purpose we will put
A^ = tan ^wJ ^~Q^,
from which comes
75tan^^ + 25tan|#=^^^^^^^^i±H=^^
denoting by a the constant
75^va+ii)^
If therefore B should be known, w could be immediately taken from the Barkerian
table containing the true anomaly to which answers the mean motion -^ ; A will
be deduced from w by means of the formula
A= ^ tan^ h w,
denoting the constant
Now, although B may be finally known from A by means of our auxiliary table,
nevertheless it can be foreseen, owing to its dijBfering so little from unity, that if
the divisor B were wholly neglected from the beginning, w and A would be
affected with a slight error only. Therefore, we will first determine roughly w
and A, putting B = l', with the approximate value of A, we will find B in our
45 RELATIONS PERTAINING SIMPLY [BoOK I.
auxiliary table, witli which we will repeat more exactly the same calculation ;
most frequently, precisely the same value of B that had been found from the
approximate value of A will correspond to the value of A thus corrected, so that a
second repetition of the operation would be superfluous, those cases excepted in
which the value of ^ may have been very considerable.
Finally, it is hardly necessary to observe that, if the approximate value of B
should in any other way whatever be known from the beginning, (which may
always occur, when of several places to be computed, not very distant from each
other, -some few are already obtained,) it is better to make use of this at once in
the first approximation : in this manner the expert computer will very often not
have occasion for even a single repetition. "We have arrived at this most rapid
approximation from the fact that B differs from unity, only by a difference of the
fourth order, and is multiplied by a very small numerical coefficient, which advan-
tage, as will now be perceived, was secured by the introduction of the quantities
E — sin E, j^Q E -j- -^^ sin E, in the place of E and sin E.
40.
Since, for the third operation, that is, the determination of the true anomaly,
the angle E is not required, but the tan ^ E only, or rather the log tan h E, that
operation could be conveniently joined with the second, provided our table sup-
plied directly the logarithm of the quantity
taxi\E
which differs from unity by a quantity of the second order. We have preferred,
however, to arrange our table in a somewhat different manner, by which, not-
withstanding the small extension, we have obtained a much more convenient
interpolation. By writing, for the sake of brevity, T instead of the tan^ h E, the
value oi A, given in article 37,
l5{E—smE)
is easily changed to
T—iT^-\-^T^—^ r*-f|f r« — etc.
A =
-T^ T+^^ T-'-^ T' + ^^ r^-etc'
Sect. 1.] to position in the orbit. 4Y
in which the law of progression is obvious. Hence is deduced, by the inversion
of the series,
^ = 1 - 1 ^ + if 5 ^^ + ^f ^ ^^ + ^Hlf 5 ^' + TsWiVi^ ^' + etc.
Putting, therefore,
C will be a quantity of the fourth order, which being included in our table, we
can pass directly to v from A by means of the formula,
, . llA-e I A rtan-|w
te°*'' = \/T^\/l-M+0 = V(i-M+g)'
denoting by y the constant
/5 + 5e
In this way we gain at the same time a very convenient computation for the
radius vector. It becomes, in fact, (article 8, YL),
gcosH^ _ q _ (l-f^-f<7)g
cos^iw (l + rjcos^if (i-j-i.J+(7)cos2iv*
41.
Nothing now remains but to reduce the inverse problem also, that is, the
determination of the time from the true anomaly, to a more expeditious form of
computation : for this purpose we have added to our table a new column for T.
T, therefore, will be computed first from v by means of the formula
1 -\-e '
then A and log B are taken from our table with the argument T, or, (which is
more accurate, and even more convenient also), G and log B, and hence A by
the formula
(1+0) r.
finally t is derived from A and B by formula [1], article 37. If it is desired to
call into use the Barkerian table here also, which however in this inveise problem
48 RELATIONS PERTAINING SDIPLY [BoOK I.
has less effect in facilitating the calculation, it is not necessary to pay any regard
to A, but we have at once
ta.n^w=z tan i v \J j^^f^,
and hence the time t, by multiplying the mean motion corresponding to the true
anomaly, tv. in the Barkerian table, by — .
42.
We have constructed with sufficient fulness a table, such as we have just
described, and have added it to this work, (Table L). Only the first part pertains
to the ellipse ; we will explain, further on, the other part, which includes the
hyperbolic motion. The argument of the table, which is the quantity A, proceeds
by single thousandths from 0 to 0.300 ; the log B and C follow, which quantities
it must be understood are given in ten millionths, or to seven places of decimals,
the ciphers preceding the significant figures being suppressed ; lastly, the fourth
column gives the quantity T computed first to five, then to six figures, which
degree of accuracy is quite sufficient, since this column is only needed to get the
values of log B and G corresponding to the argument T, whenever t is to be
determined from v by the precept of the preceding article. As the inverse prob-
lem which is much more frequently employed, that is, the determination of v and
r from i, is solved altogether without the help of T, we have preferred the quan-
tity A for the argument of our table rather than T, which would otherwise have
been an almost equally suitable argument, and would even have facilitated a little
the construction of the table. It will not be unnecessary to mention, that all the
numbers of the table have been calculated from the beginning to ten places, and
that, therefore, the seven places of figures which we give can be safely relied upon;
but we cannot dwell here upon the analytical methods used for this work, by a
full explanation of which we should be too much diverted from our j^lan.
Finally, the extent of the table is abundantly sufficient for all cases in which it
is advantageous to pursue the method just explained, since beyond the limit
^ = 0.3, to which answers r= 0.392374, or ^=64° 7', we may, as has been
shown before, conveniently dispense with artificial methods.
Sect. 1.]
TO POSITION EN THE ORBIT.
49
43.
We add, for the better illustration of the preceding investigations, an example
of the complete calculation for the true anomaly and radius vector from the time,
for which purpose we will resume the numbers in article 38. We put then e =
0.9674567, log ^=: 9.7656500, ^5=: 63.54400, whence, we first derive the constants
log a = 0.03052357, log (i = 8.2217364, log y = 0.0028755.
Hence we have log at= 2.1083102, to which corresponds in Barker's table
the approximate value of w = 99° 6' whence is obtained A= 0.022926, and from
our table log B = 0.0000040. Hence, the correct argument with which Barker's
table must be entered, becomes log-^ = 2.1083062, to which answers to = 99° 6'
13".14 ; after this, the subsequent calculation is as follows : —
logtanH^; . . . 0.1385934
log fi 8.2217364
log J. 8.3603298
A= 0.02292608
log tan iw 0.0692967
log 7 0.0028755
i Comp. log(l— iA-{-C). 0.0040143
log tan H' 0.0761865
hence log B in the same manner as before ; i v= 50° 0' 0"
0=
iA^C=
0.0000242
0.9816833
1.0046094
log^
2 Comp. log cos I V
log(l-|^+^. .
c.iog(i4-i^4-c^.
100 0 0
9.7656500
0.3838650
9.9919714
9.9980028
logr 0.1394892
If the factor B had been wholly neglected in this calculation, the true anomaly
would have come out affected with a very slight error (in excess) of 0".l only.
' 44.
It will be in our power to despatch the hyperbolic motion the more briefly,
because it is to be treated in a manner precisely analogous to that which we
have thus far expounded for the elliptic motion.
7
50 RELATIONS PERTAINING SIMPLY [BoOK 1.
We present the equation between the time t and the auxiliary quantity u in
the following form : —
in which the logarithms are hyperbolic, and
2V(^-^) + Aiog^^
is a quantity of the first order,
a quantity of the third order, when log u may be considered as a small quantity
of the first order. Putting, therefore,
A will be a quantity of the second order, but B will differ from unity by a differ-
ence of the fourth order. Our equation will then assume the following form : —
B{2{e-l)A^-^r^A'^-\-^e)J^)=M{^^f [2]
which is entirely analogous to equation [1] of article 37. Putting moreover,
T will be a quantity of the second order, and by the method of infinite series
will be found
Wherefore, putting
C will be a quantity of the fourth order, and
^— i_|2' •
Finally, for the radius vector, there readily follows from equation VII., article 21,
(1 — T)CQS'^V~{1 — ^A^0)C06^^V'
Sect. 1.] to position in the orbit. 51
45.
The latter part of the table annexed to this work belongs, as we have remarked
above, to the hyperbolic motion, and gives for the argument A (common to both
parts of the table), the logarithm of B and the quantity 0 to seven places of
decimals, (the preceding ciphers being omitted), and the quantity T to five and
afterwards to six figures. The latter part is extended in the same manner as
the former to ^=0.300, corresponding to which is r= 0.241207, m= 2.930,
or = 0.341, i^=+52°19'; to extend it further would have been superfluous,
(article 36).
The following is the arrangement of the calculation, not only for the determi-
nation of the time from the true anomaly, but for the determination of the true
anomaly from the time. In the former problem, T will be got by means of the
formula
^=^-^tanHe^;
e-\- i
with T our table will give log B and C, whence will follow
finally t is then found from the formula [2] of the preceding article. In the last
problem, will first be computed, the logarithms of the constants
5^-5
f^ l+9e
' — V l+9e*
A will then be determined from t exactly in the same manner as in the elliptic
motion, so that in fact the true anomaly w may correspond in Barker's table to
the mean motion -^,and that we may have
A= ^ tan^ i w ;
the approximate value of A will be of course first obtained, the factor B being
52 EELATIONS PERTATNING SIMPLY [BoOK I.
either neglected, or, if the means are at hand, being estimated ; our table will
then furnish the approximate value of B, with which the work will be repeated ;
the new value of B resulting in this manner will scarcely ever suffer sensible cor-
rection, and thus a second repetition of the calculation will not be necessary. C
will be taken from the table with the corrected value of A, which being done we
shall have,
, , ytanjw {I -\- ^ A -\- G)q
^^^*^""V(l+^^+^)' ^— (l-i-4+0)cos^i^;•
From this it is evident, that no difference can be perceived between the formulas
for elliptic and hyperbolic motions, provided that we consider §, A, and T, in the
hyperbolic motion as negative quantities.
46.
It will not be unprofitable to elucidate the hyperbolic motion also by some
examples, for which purpose we will resume the numbers in articles 23, 26.
I. The data are e = 1.2618820, log ^ = 0.0201657, y=18°5r0'': ^ is
required. We have
2 log tan I y . . . . 8.4402018 log 7^ 7.5038375
log^ 9.0636357 l<^g(lH-^)- • • 0.0000002
'+^ C.log(l — 1^) . 0.0011099
1^^^ • ^-^OSS^^^ log^ 7.5049476
T:=z 0.00319034
log^= 0.0000001
0= 0.0000005
l^g^Tli • • • 2-2^^^^^^ ^^^'-^F^i-drS ■ ' • 2.8843582
log^* 8.7524738 log^* 6.2574214
log 13.77584 = . . 1.1391182 log 0.138605 .=: 9.1417796.
0.13861
13.91445 = ^;.
II. e and g remaining as before, there is given i! = 65.41236; v and r are
required. We find the logarithms of the constants,
Sect. 1.]
TO POSITION IN THE ORBIT.
53
log a = 9.9758345
log /? = 9.0251649
log y = 9.9807646.
Next we have log azf = 1.7914943, whence by Barker's table the approxhuate
value of e^=70°3r44", and hence ^ = 0.052983. To this A in our table
answers log B = 0.0000207 ; from which, log ^ = 1.7914736, and the corrected
value of w^ 70°3r36".86. The remaining operations of the calculation are as
follows : —
log tan ^ et- 9.8494699
logy 9.9807646
iC.log(l + |^4-C) . 9.9909602
2 log tan Im; . . . 9.6989398
log /? 9.0251649
log^ 8.7241047
A= . . . . . . 0.05297911
log B as before,
C= . . 0.0001252
l^AA-\-C= . . 1.0425085
1 — |iL+C= . . 0.9895294
log tan iy 9.8211947
iv=z ... 33°3r30''.02
v= ... 67 3 0 .04
log^ 0.0201657
2 Clog cos ^y . . . . 0.1580378
log (1 + 1^4- (7) . . 0.0180796
C.log(l — ■1^4-C') . . 0.0045713
logr 0.2008544
Those which we found above (article 26), y = 67°2'69".78, logr = 0.2008541,
are less exact, and v should properly have resulted = 67° 3' 0".00, with which
assumed value, the value of ^ had been computed by means of the larger tables.
SECOND SECTION.
RELATIONS PERTAINING SIMPLY TO POSITION IN SPACE.
47.
In the first section, the motion of heavenly bodies in their orbits is treated
without regard to the position of these orbits in space. For determining this
position, by which the relation of the places of the heavenly body to any other
point of space can be assigned, there is manifestly required, not only the position
of the plane in which the orbit lies with reference to a certain known plane (as,
for example, the plane of the orbit of the earth, the ecliptic), but also the position
of the apsides in that plane. Since these things may be referred, most advanta-
geously, to spherical trigonometry, we conceive a spherical surface described
with an arbitrary radius, about the sun as a centre, on which any plane passing
through the sun will mark a great circle, and any right line drawn from the
sun, a point. For planes and right lines not passing through the sun, we draw
through the sun parallel planes and right lines, and we conceive the great circles
and points in the surface of the sphere corresponding to the latter to represent
the former. The sphere may also be supposed to be described with a radius
infinitely great, in which parallel planes, and also parallel right lines, are repre-
sented in the same manner.
Except, therefore, the plane of the orbit coincide with the plane of the ecliptic,
the great circles corresponding to those planes (which we will simply call the orbit
and the ecliptic) cut each other in two points, which are called nodes ; in one of
these nodes, the body, seen from the sun, will pass from the southern, through the
ecliptic, to the northern hemisphere, in the other, it will return from the latter to
the former ; the former is called the ascending, the latter the descending node. We
(54)
Sect. 2.] TO position in space. 55 '
fix the positions of the nodes in the ecliptic by means of their distance from the
mean vernal equinox [longitude) counted in the order of the signs. Let, in fig. 1,
S2 be the ascending node, AQ> B part of the ecliptic, G Q, D part of the orbit ;
let the motions of thg earth and of the heavenly body be in the directions from A
towards B and from C towards D, it is evident that the spherical angle which 9> D
makes with 9, B can increase from 0 to 180°, but not beyond, without 9, ceasing
to be the ascending node : this angle we call the inclination of the orbit to the
ecliptic. The situation of the plane of the orbit being determined by the longi-
tude of the node and the inclination of the orbit, nothing further is wanted
except the distance of the perihelion from the ascending node, which we reckon
in the direction of the motion, and therefore regard it as negative, or between
180° and 360°, whenever the perihelion is south of the ecliptic. The following
expressions are yet to be observed. The longitude of any j)oint whatever in
the circle of the orbit is counted from that point which is distant just so far back
from the ascending node in the orbit as the vernal equinox is back from the same
point in the ecliptic : hence, the longitude of the perihelion will be the sum of the
longitude of the node and the distance of the perihelion from the node ; also, the
true longitude in oriit of the body will be the sum of the true anomaly and the
longitude of the perihelion. Lastly, the sum of the mean anomaly and longitude
of the perihelion is called the mean longitude : this last expression can evidently
only occur in elliptic orbits.
48.
In order, therefore, to be able to assign the place of a heavenly body in space
for any moment of time, the following things must be known.
I. The mean longitude for any moment of time taken at will, which is called
the epoch : sometimes the longitude itself is designated by the same name. For
the most part, the beginning of some year is selected for the epoch, namely, noon
of January 1 in the bissextile year, or noon of December 31 preceding, in the
common year.
IL The mean motion in a certain interval of time, for example, in one mean
solar day, or in 365, 365^, or 36525 days.
56 RELATIONS PERTAESriNG SIMPLY [BoOK I.
III. The semi-axis major, whicli indeed might be omitted when the mass of
the body is known or can be neglected, since it is already given by the mean
motion, (article 7) ; both, nevertheless, are usually given for the sake of con-
venience.
IV. Eccentricity. V. Longitude of the perihelion. VI. Longitude of the
ascending node. VII. Inclination of the orbit.
These seven things are called the elements of the motion of the body.
In the parabola and hyperbola, the time of passage through the perihelion
serves in place of the first element ; instead of II., are given what in these
species of conic sections are analogous to the mean daily motion, (see article
19 ; in the hyperbolic motion the quantity X Jcb~^, article 23). In the hyperbola,
the remaining elements may be retained the same, but in the parabola, where
the major axis is infinite and the eccentricity = 1, the perihelion distance alone
will be given in place of the elements III. and IV.
49.
According to the common mode of speaking, the inclination of the orbit,
which we count from 0 to 180°, is only extended to 90°, and if the angle made
by the orbit with the arc Q, B exceeds a right angle, the angle of the orbit with
the arc 9> A, which is its complement to 180°, is regarded as the inclination of
the orbit ; in this case then it will be necessary to add that the motion is retrograde
(as if, in our fiigure, E Q, F should represent a part of the orbit), in order that it
may be distinguished from the other case where the motion is called direct. The
longitude in orbit is then usually so reckoned that in Q, it may agree with the
longitude of this point in the ecliptic, but decrease in the direction Q, F; the initial
point, therefore, from which longitudes are counted contrary to the order of
motion in the direction ^ F, is just so far distant from Q,, as the vernal equinox
from the same Q, in the direction Q, A. Wherefore, in this case the longitude of
the perihelion will be the longitude of the node diminished by the distance of
the perihelion from the node. In this way either form of expression is easily con-
verted into the other, but we have preferred our own, for the reason that we
might do away with the distinction between the direct and retrograde motion,
Sect. 2.] to position in space. 57
and use always the same formulas for both, while the common form may fre-
quently require double precepts.
50.
The most simple method of determining the position, with respect to the
ecliptic, of any point whatever on the surface of the celestial sphere, is by means
of its distance from the ecliptic {latitude), and the distance from the equinox of
the point at which the ecliptic is cut by a perpendicular let fall upon it, {longi-
tude). The latitude, counted both ways from the ecliptic up to 90°, is regarded as
positive in the northern hemisphere, and as negative in the southern. Let the
longitude X, and the latitude /5, correspond to the heliocentric place of a celestial
body, that is, to the projection upon the celestial sphere of a right line drawn
from the sun to the body ; let, also, u be the distance of the heliocentric place
from the ascending node (which is called the argument of the latitude), i be the
inclination of the orbit, S2 the longitude of the ascending node; there will exist
between i, u, (i, X — Q, which quantities will be parts of a right-angled spherical
triangle, the following relations, which, it is easily shown, hold good without any
restriction : —
I. tan (X — 9,) = cos ^ tan u
11. tan fi = tan^ sin (X — Q)
III. sin /i = sin i sin u
IV. cos u = cos (3 cos (X — Q).
When the quantities i and u are given, X — Q will be determined from them by
means of equation I, and afterwards /5 by II. or by III., if /? does not approach
too near to + 90° ; formula IV. can be used at pleasure for confirming the cal-
culation. Formulas I. and IV. show, moreover, that X — Q and u always lie in
the same quadrant when ^ is between 0° and 90° ; X — Q, and 360° — u, on the
other hand, will belong to the same quadrant when ^ is between 90° and 180°, or,
according to the common usage, when the motion is retrograde : hence the ambi-
guity which remains in the determination of A, — Q by means of the tangent
according to formula I., is readily removed.
58 RELATIONS PERTAIMNG SEVITLT • [BoOK I.
The following formulas are easily deduced from the combination of the pre-
ceding : —
V. sm{u — X -}- S ) = 2 sin^ I i sin it cos {I — Q )
VI. sin (t( — 2. -|- a ) = tan i i sin /? cos {X — Q)
VII. sin (it — X -|- S^ ) = tan i i tan /:? cos u
VIII. sin (tf + X — ^ ) = 2 cos^ I i sin u cos {I — 9,)
IX. sin (m -|- >. — S2 ) = cotan h i sin /? cos (^ — 9,)
X. sin (it -|- X — 0,)^= cotan ^ i tan /? cos m.
The angle u — 1^9>, v;hen i is less than 90°, or u-\-l — 9, when i is more
than 90°, called, according to common usage, the reduction to the ecliptic, is, in fact,
the difference between the heliocentric longitude I and the longitude in orbit,
which last is by the former usage g^ + w, by ours Q> -j- u. When the inclination
is small or differs but little from 180°, the same reduction may be regarded as a
quantity of the second order, and in this case it will be better to compute first ^
by the formula III., and afterwards I by VII. or X., by which means a greater
precision will be attained than by formula I.
If a perpendicular is let fall from the place of the heavenly body in space
upon the plane of the ecliptic, the distance of the point of intersection from the
sun is called the curtate distance. Designating this by /, the radius vector likewise
by r, we shall have
XL / = rcos|?.
51.
As an example, we will continue further the calculations commenced in arti-
cles 13 and 14, the numbers of which the planet Juno furnished. We had
found above, the true anomaly 315°r23".02, the logarithm of the radius vector
0.3259877: now let « = 13°6'44".10, the distance of the perihelion from the
node =241°10'20".57, and consequently « = 196°ir43".59 ; finally let 9 =
171° 7'48".73. Hence we have : —
logtauM .... 9.4630573 log sin (X— g2) . • . . 9.4348691n
log cose .... 9.9885266 log tan e 9.3672305
loglan {1—9). . 9.4515839 log tan /S 8.8020996 «
Sect. 2.]
I— S
TO POSITION IN SPACE.
59
Q, = 195°4r40".25
1= 6 55 28.98
logr ..... . 0.3259877
log cos /5 9.9991289
log / 0.3251166 log cos
/?= — 3°3r40':02
log cos ^ 9.9991289
loffcosX— a ... 9.9832852 »
9.9824141%
9.9824141%.
The calculation by means of formulas III, VII. would be as foUows : —
log sin ?{ . . ., . 9.4454714%
log sin « 9.3557570
log sin /5 .... 8.8012284%
/?= — 3°3r40".02
log tan i 2 9.0604259
log tan ^ 8.8020995%
log cos M 9.9824141%
log
sin (u —
X-i-Q)
. 7.8449395
u —
-l-\-Q
=
0°24' 8".34
X—Q
=
195 47 40 .25.
52.
Regarding ^ and u as variable quantities, the differentiation of equation III.,
article 50, gives
cotan /3 d/5 = cotan ^d^' -|- cotan udu,
or
XII. d|3 = sin(^ — ^)d^-|- sin ^ cos (X — 9,)du.
In the same manner, by differentiation of equation I. we get
XIII. d(X— Q) = — tiin(Uos{l— Q)di-{-^du.
Finally, from the differentiation of equation XL comes
dr = cos /? dr — r sin/3d/5,
XIV. dr = cosjSdr — r sin ^ sin(X — Q)di — rsm^ sin ^ cos (I — Q)du.
In this last equation, either the parts that contain di and du are to be divided by
206265", or the remaining ones are to be multiplied by this number, if the
changes of ^' and u are supposed to be expressed in minutes and seconds.
60 RELATIONS PERTAINESIG SBIPLY [BoOK I.
53.
The position of any point whatever in space is most conveniently deter-
mined by means of its distances from three planes cutting each other at right
angles. Assuming the plane of the ecliptic to be one of these planes, and denot-
ing the distance of the heavenly body from this plane by 2", taken positively on
the north side, negatively on the south, we shall evidently have s = /tan /:J =:
r sin /5 ^ r sin i sin u. The two remaining planes, which we also shall consider
drawn through the sun, will project great circles upon the celestial sphere, which
will cut the ecliptic at right angles, and the poles of which, therefore, will lie in
the ecliptic, and will be at the distance of 90° from each other. We call that pole
of each plane, lying on the side from which the positive distances are counted,
the positive pole. Let, accordingly, iV and N -\- 90° be the longitudes of the
positive poles, and let distances from the planes to which they respectively
belong be denoted by x and i/. Then it will be readily perceived that we have
a; = r'cos {X — N)
= r cos {^} cos (A, — Q,) cos [N — 9,)-\-r cos ^i sin (X — ^) sin {N — Q, )
y = / sin (A, — iV)
= r cos fi sin (^ — 9>) cos (iV — 9>) — r cos (i cos {I — ^) sin (iV — Q ),
which values are transformed into
x = r cos {jV — Q, ) cos u -|- r cos i sin (iV — Q, ) sin u
2/ = r cos ^' cos {JV — 9,) sin u — rsin (iV — Q, ) cosu.
If now the positive pole of the plane of x is placed in the ascending node, so that
iV^= S2, we shall have the most simple expressions of the coordinates x,f/, z, —
a; = r cos u
y = r cos i sin u
s = r sin i sin u .
But, if this supposed condition does not occur, the formulas given above will
still acquire a form almost equally convenient, by the introduction of four
auxiliar}'^ quantities, a, h, A, B, so determined as to have
Sect. 2,] TO POSITION m space. 61
cos {JV — 9,) = a sin A
cos i sin (iV — S^ ) = a cos A
— sin {J\^— 9,) = bsmB
cos i cos (iV — Q)=:ib cos B,
(see article 14, II.). We shall then evidently have
X =^ra sm{u -\- A)
2/ = ri sin (u-\- B)
g = r sin i sin u .
54.
The relations of the motion to the ecliptic explained in the preceding article,
will evidently hold equally good, even if some other plane should be substituted
for the ecliptic, provided, only, the position of the plane of the orbit in respect
to this plane be known ; but in this case the expressions longitude and latitude
must be suppressed. The problem, therefore, presents" itself: From the known
position of the plane of the orbit and of another new plane in respect to the ecliptic, to
derive the position of the plane of the orbit in respect to the neiv plane. Let nQ>, 9>9>',
nO,' he parts of the great circles which the plane of the ecliptic, the plane of the
orbit, and the new plane, project upon the celestial sphere, (fig. 2). In order
that it may be possible to assign, without ambiguity, the inclination of the second
circle to the third, and the place of the ascending node, one direction or the other
must be chosen in the third circle, analogous, as it were, to that in the ecliptic
which is in the order of the signs; let this direction in our figure be from n toward
9>'. Moreover, of the two hemispheres, separated by the circle nO,', it will be
necessary to regard one as analogous to the northern hemisphere, the other to
the southern ; these hemispheres, in fact, are already distinct in themselves, since
that is always regarded as the northern, which is on the right hand to one moving
forward* in the circle according to the order of the signs. In our figure, then, Q,,
n, Q,', are the ascending nodes of the second circle upon the first, the third upon
the first, the second upon the third; 180° — nQQ>', Q, n Q,' , n Q>' Q, the inclina-
* In the inner surface, ttat is to say, of the sphere represented by our figure.
62 RELATIONS PERTAINING SEVIPLY [BouR 1.
tions of the second to the first, the third to the first, the second to the third.
Our problem, therefore, depends upon the solution of a spherical triangle, in
which, from one side and the adjacent angles, the other parts are to be deduced.
We omit, as sufficiently well known, the common precepts for this case given
in spherical trigonometry : another method, derived from certain equations, which
are sought in vain in our works on trigonometry, is more conveniently emploj'ed.
The following are these equations, which we shall make frequent use of in future:
a, h, c, denote the sides of the spherical triangle, and A, B, C, the angles oppo-
site to them respectively : —
I.
&m\{b — c) sin ^ {B— G)
sin -I a cos ^ A
TT ?i? 2" (^ ~l~ ^) ^°^ 2 (-^ — ^)
-,jy COS ^{h — c) &m^{B-\-G)
coi\a cos|^^
j-y- cos 1(5 + c) cos \ {B-\- O)
Although it is necessary, for the sake of brevity, to omit here the demonstration
of these propositions, any one can easily verify them in Iriangles of which neither
the sides nor the angles exceed 180°. But if the idea of the spherical triangle is
conceived in its greatest generality, so that neither the sides nor the angles are
confined within any limits whatever (which affords several remarkable advan-
tages, but requires certain preliminary explanations), cases may exist in which it
is necessary to change the signs in all the preceding equations ; since the former
signs are evidently restored as soon as one of the angles or one of the sides is
increased or diminished 360°, it will always be safe to retain the signs as we
have given them, whether the remaining parts are to be determined from a side
and the adjacent angles, or from an angle and the adjacent sides ; for, either
the values of the quantities sought, or those differing by 360° from the true val-
ues, and, therefore, equivalent to them, will be obtained by our formulas. We
reserve for another occasion a fuller elucidation of this subject : because, in the
meantime, it will not be difficult, by a rigorous induction, that is, by a complete
enumeration of all the cases, to prove, that the precepts which we shall base upon
Sect. 2.] TO position in space. 63
these formulas, both for the solution of our present problem, and for other pur-
poses, hold good in all cases generally.
55.
Designating as above, the longitude of the ascending node of the orbit upon
the ecliptic by Q, the inclination by ^'; also, the longitude of the ascending node
of the new plane upon the ecliptic by n, the inclination by e ; the distance of the
ascending node of the orbit upon the new plane from the ascending node of the
new plane upon the ecliptic (the arc n^' in fig. 2) by 9,', the inclination of the
orbit to the new plane by i' ; finally, the arc from 9, to ^' in the direction of the
motion by J : the sides of our spherical triangle will be Q, — n, Q,', J, and the
opposite angles,/, 180° — /, e. Hence, according to the formulas of the preceding
article, we shall have
sin ^ i' sin ^ (9,' -\- J) =: sin i [Q — n) sin J («' -|- e)
sin i i'cos ^ [9' -\- J) = cos i (9, — n) sin i [i — e)
cos hi' sin h{9' — ^) = sin k {9 — oi) cos h (^' -|- e)
cos|^''cos t {9' — z/) =cos ^ (S2 — n)co8h {i — £).
The two first equations will furnish h [9' -\- J) and sin k i' ; the remaining two,
h{9' — z/) and cos ^^''; from ^ {9' -\- J) and h {9' — z/) will follow S^'and//;
from sin i ^' and cos ^ i' (the agreement of which will serve to prove the calcula-
tion) will result i'. The uncertainty, whether h{9' -{- /I) and i {9' — /I) should
be taken between 0 and 180° or between 180° and 360°, will be removed in this
manner, that both sin i i\ cos \ i\ are positive, since, from the nature of the case, i'
must fall below 180°.
56.
It will not prove unprofitable to illustrate the preceding precepts by an
example. Let g2 == 172°28'13".7, e= 34° 38' I'M; let also the new plane be
parallel to the equator, so that n = 180° ; we put the angle e, which will be the
obliquity of the ecliptic = 23° 27' 55".8. We have, therefore.
64
RELATIONS PERTAINING SIMPLY
[Book 1.
U 71 =
^ -|- £ :=
log sin i (S2 — n) . .
log sin i (/-)- e) . . .
log cos 2 («' -f- ^) • • •
, .; c V, ■ have
log sin i i' sin k {9>' -\- J)
log sin h I cos -^ { S ' -j- '^)
-7°3r46".3
58 5 56 .9
11 10 5 .3
. 8.8173026 ?z
. 9.6862484
. 9.9416108
8.5035510«
8.9872023
^(Q_j?)= _3'
H« + ^)= 29
|(^ — e)= 5
log cos ^ ( S? — n) . .
logsin|(^' — e) . . .
log cos \{i — £) . . .
log cos h i' sin I ( S^ ' — /t)
log cos hi' COS h {Q> ' — //)
45'53".15
2 58 .45
35 2.65
9.9990618
8.9881405
9.9979342.
8.7589134^2
9.9969960
whence I ( 9,'-\- A) = 341° 49' 19".01 whence h{9.' — A) = 356° 41' 31".43
log sin ^e" 9.0094368 log cos ^/ 9.9977202.
Thus we obtain ^ ^•' = 5° 51' 56".445, ^" = 11° 43'52".89, S^'= 338° 30' 50".43,
A = — 14° 52' 12".42. Finally, the point ?? evidently corresponds in the celestial
sphere to the autumnal equinox ; for which reason, the distance of the ascending
node of the orbit on the equator from the vernal equinox (its ri(/ht ascetision)
will be 158°30'50".43.
In order to illustrate article 53, we will continue this example still further,
and will develop the formulas for the coordinates with reference to the three
planes passing through the sun, of which, let one be parallel to the equator, and
let the positive poles of the two others be situated in right ascension 0° and 90° :
let the distances from these planes be respectively z, x, y. If now, moreover,
the distances of the heliocentric place in the celestial sphere from the points 9,,
S ', are denoted respectively by «, li, we shall have ?«'= « — A=iu-\- 14° 52' 12".42,
and the quantities which in article 53 were represented by ^, N — 9> , ^<, will here
be /, 180° — S^', iL Thus, from the formulas there given, follow,
log a sin .4 . . . . 9.9687197?^ log^sin^ . . . . 9.5638058
log a cos ^ . . . . 9.5546380;? log^cos^ . . . . 9.9595519 ?z
whence ^ = 248°55'22".97
log a 9.9987923
We have therefore,
whence^ = 158° 5' 54".97
log* 9.9920848.
Sect. 2.] to position in space. 65
X = ar sin {i/-{-24:S° 6b' 22". 97) = ar sin (« + 263°47'3.5".39)
t/ = br sin (^f' -[- 158 5 54 .97) = br sin {u + 172 58 7 .39)
2r = crsinz/ = cr sin (^J -[- 14 52 12.42)
in which log c = log sin i' = 9.3081870.
Another solution of the problem here treated is found in Vo7i Zach's MonatUche
Correspondens, B. IX. p. 385.
57.
Accordingly, the distance of- a heavenly body from any plane passing through
the sun can be reduced to the form krmi [v -|- K), v denoting the true anomaly;
Jc will be the sine of the inclination of the orbit to this plane, K the distance
of the perihelion from the ascending node of the orbit in the same plane. So far
as the position of the plane of the orbit, and of the line of apsides in it, and also
the position of the plane to which the distances are referred, can be regarded as
constant, Jc and K will also be constant. In such a case, however, that method
will be more frequently called into use in which the third assumption, at least, is
not allowed, even if the perturbations should be neglected, which always affect
the first and second to a certain extent. This happens as often as the distances
are referred to the equator, or to a plane cutting the equator at a right angle
in given right ascension: for since the position of the equator is variable, owing to
the precession of the equinoxes and moreover to the nutation (if the true and not
the mean position should be in question), in this case also k and K will be subject
to changes, though undoubtedly slow. The computation of these changes can be
nlade by means of differential formulas obtained without difficulty: but here
it may be, for the sake of brevity, sufficient to add the differential variations
oii', Q,' and J, so far as they depend upon the changes of Q, — n and £.
d/ = sin £ sin Q,' A [Q, — n) — cos S^'de
T -^ , sin i cos A
d(S^— n)-f-
tarn
sin '
^^^Mn.cu.,6 d(S^— n)4---^iL^d£.
sint ^ '^ ' sln^
Finally, when the problem only is, that several places of a celestial body with
9
66 RELATIONS PERTAKnNG SIMPLY [BoOK 1.
respect to such variable planes may be computed, which places embrace a mod-
erate interval of time (one year, for example), it will generally be most con-
venient to calculate the quantities «, A, b, B, c, C, for the two epochs between
which they fall, and to derive from them by simple interpolation the changes for
the particular times proposed.
58.
Our formulas for distances from given planes involve v andr; when it is
necessary to determine these quantities first from the time, it will be possible to
abridge part of the operations still more, and thus greatly to lighten the labor.
These distances can be immediately derived, by means of a very simple formula,
from the eccentric anomaly in the ellipse, or from the auxiliary quantity F or ?f
in the hyperbola, so that there will be no need of the computation of the true
anomaly and radius vector. The expression kr sin {v -\- K) is changed ;
I. For the ellipse, the symbols in article 8 being retained, into
a^cosg) cos ^ sin ^-|- a^sin^(cos^ — e).
Determining, therefore, I, L, X, by means of the equations
aJcsm K^=lsinL
ak cos(p cosK=lcosL
— eak smK= — el smL=zX,
our expression passes into Ism [U -\- Z) -{- X, in which /, X, X will be constant, so
far as it is admissible to regard /c, K, e as constant ; but if not, the same precepts
which we laid down in the preceding article will be sufficient for computing thei)
changes.
We add, for the sake of an example, the transformation of the expression for
X found in article 56, in which we put the longitude of the perihelion = 121° 17'
34".4, 9) = 14" 13'3r.97, log a = 0.4423790. The distance of the perihelion from
the ascending node in the ecliptic, therefore, = 308° 49' 20".7 = ^( — v; hence
K= 212°36'56".09. Thus we have.
Sect. 2.] TO position in space. 67
logaJc ..... 0.4411713 log/sinZ .... 0.1727000^2
logsinX .... 9.7315887^^ log/cosX .... 0.3531154;?
log«7fcos9 . . . 0.4276456 whence L = 213°25'5r.30
log cos ^ .... 9.9254698 ?2. log/=: 0.4316627
log;ir= 9.5632352
1= +0.3657929.
n. In the hyperbola the formula ^r sin (y -|-^), by article 21, passes into
X -}- |U. tan F-\-v sec F, if we put eb k sin K= I, ik tan i/^ cos ir= ^i, — bk sinK
=:v ; it is also, evidently, allowable to bring the same expression under the form
cosF
If the auxiliary quantity u is used in the place of F, the expression ^-r sin (v-\-S)
will pass, by article 21, into
in which a, (i, y, are determined by means of the formulas
a = 1 = ehk sin K
(j = ^v -\- fi,) =— i ehk sin {K—xi))
y ^ ^ {y — /^) = — h ebk sm. {^-\- <f )•
III. In the parabola, where the true anomaly is derived directly from the time,
nothing would remain but to substitute for the radius vector its value. Thus,
denoting the perihelion distance by q, the expression A^r sin {v -\- K) becomes
59.
The precepts for determining distances from planes passing through the sun
may, it is evident, be applied to distances from the earth ; here, indeed, only the
most simple cases usually occur. Let R be the distance of the earth from the sun,
L the heliocentric longitude of the earth (which differs 180° from the geocentric
longitude of the sun), lastly, X, Y, Z, the distances of the earth from three planes
cuttino; each other in the sun at riw:ht angles. Now if
68
RELATIONS PERTAINING SEMPLY
[Book I.
I. The plane of Z is the ecliptic itself, and the longitudes of the poles of the
remaining planes, the distances from which are X, F, are respectively N, and
i\r^ 90°; then
X^i?cos(Z — i\^), r = i?sin(X — i\^), ^=0.
n. If the plane of Z is parallel to the equator, and the right ascensions of the
poles of the remaining planes, from which the distances are X., Y, are respectively
0° and 90°, we shall have, denoting by £ the obliquity of the ecliptic,
X=BgosL, Y :^ H cos e sin L, Z= E sine sin L.
The editors of the most recent solar tables, the illustrious Von Zach and de
Lambke, first began to take account of the latitude of the sun, which, produced
by the perturbations of the other planets and of the moon, can scarcely amount
to one second. Denoting by B the hehocentric latitude of the earth, which will
always be equal to the latitude of the sun but affected with the opposite sign, we
shall have,
In Case I.
X=BgosBcos{L — JY)
Y=EcosBsin{L—JY)
Z=E sin B
In Case 11.
X^= E cos B cos L
Y=z E cos B cos e sin X — E sin B sin £
Z = E cos B sin esinL-\- E sin B cos e.
It will always be safe to substitute 1 for cos B, and the angle expressed in parts
of the radius for sin B.
The coordinates thus found are referred to the centre of the earth. If ^, -)], C,
are the distances of any point whatever on the surface of the earth from three
planes drawn through the centre of the earth, parallel to those which were drawn
through the sun, the distances of tliis point from the planes passing through the
sun, will evidently be X-|- ^; ^-\-V^ Z -\-L: the values of the coordinates I, i], L,
are easily determined in both cases by the following method. Let () be the radius
of the terrestrial globe, (or the sine of the mean horizontal parallax of the sun,)
I the longitude of the point at which the right line drawn from the centre of the
earth to the point on the surface meets the celestial sphere, ('i the latitude of the
same point, a the right ascension, d the declination, and we shall have,
Sect. 2.] to position m space. 69
In Case I.
^ = QCOS^ COS (X JV)
?j = ^ COS jS sin (^ — JV)
C = 9 sin jS
In Case II.
^ = Q cosd COS a
rj = Q cos d sia a
This point of the celestial sphere evidently corresponds to the zenith of the
place on the surface (if the earth is regarded as a sphere), wherefore, its right
ascension agrees with the right ascension of the mid-heaven, or with the sidereal
time converted into degrees, and its declination with the elevation of the pole ;
if it should be worth while to take account of the spheroidal figure of the earth,
it would be necessary to adopt for d the connected elevation of the pole, and for
Q the true distance of the place from the centre of the earth, which are deduced
by means of known rules. The longitude and latitude I and /3 will be derived
from a and d by known rules, also to be given below : it is evident that I coin
cides with the longitude of the nonagesimal, and 90° — 13 with its altitude.
60.
If X, y, z, denote the distances of a heavenly body from three planes cutting
each other at right angles at the sun; X, Y, Z, the distances of the earth (either
of the centre or a point on the surface), it is apparent that x — X,i/ — Y, z — Z^
would be the distances of the heavenly body from three planes drawn through
the earth parallel to the former; and these distances would have the same relation
to the distance of the body from the earth and its geocentric place^ (that is, the place
of its projection in the celestial sphere, by a right hne drawn to it from the earth),
which x,t/, gjhave to its distance from the sun and the heUocentric place. Let J
be the distance of the celestial body from the earth ; suppose a perpendicular in
the celestial sphere let fall from the geocentric place on the great circle which
corresponds to the plane of the distances z, and let a be the distance of the
intersection from the positive pole of the great circle which corresponds to the
* In the broader sense : for properly this expression refers to that case in which the right line is
drawn from the centre of the earth.
70 RELATIONS PERTAINING SIMPLY [BoOK 1.
plane of the distances x; and, finally, let h be the length of this perpendicular, or
the distance of the geocentric place from the great chcle corresponding to the
distances s. Then h will be the geocentric latitude or declination, according as the
plane of the distances s is the ecUptic or the equator ; on the other hand, a -j- JV
will be the geocentric longitude or right ascension, if JV denotes, in the fonner
case, the longitude, in the latter, the right ascension, of the pole of the plane of
the distances x. Wherefore, we shall have
X — X= J cos b cos a
y — F= A cos ^ sin a
z — Z ■=^ A ^va.h .
The two first equations will give a and A co^h; the latter quantity (which must
be positive) combined with the thhd equation, will give h and A.
61.
We have given, in the preceding articles, the easiest method of determining
the geocentric place of a heavenly body with respect to the ecliptic or equator,
either free from parallax or affected by it, and in the same manner, either free
from, or affected by, nutation. In what pertains to the nutation, all the difference
will depend upon this, whether we adopt the mean or true position of the equator;
as in the former case, we should count the longitudes from the mean equinox,
in the latter, from the true, just as, in the one, the mean obliquity of the ecliptic
is to be used, in the other, the true obliquity. It appears at once, that the greater
the number of abbreviations introduced into the computation of the coordinates,
the more the preliminary operations which are required ; on which account, the
superiority of the method above explained, of deriving the coordinates immedi-
ately from the eccentric anomaly, will show itself especially when it is necessary
to determine many geocentric jDlaces. But when one place only is to be com-
puted, or very few, it would not be worth while to undertake the labor of calcu-
lating so many auxiliary quantities. It will be preferable in such a case not to
depart from the common method, according to which the true anomaly and radius
vector are deduced from the eccentric anomaly; hence, the heliocentric place
Sect. 2.] to position in space. 71
with respect to tlie ecliptic ; hence, the geocentric longitude and latitude ; and
hence, finally, the right ascension and declination. Lest any thing should seem
to be wanting, we will in addition briefly explain the two last operations.
62.
Let X be the heliocentric longitude of the heavenly body, /? the latitude ; I the
geocentric longitude, h the latitude, r the distance from the sun, J the distance
from the earth ; lastly, let L be the heliocentric longitude of the earth, B the lat-
itude, R its distance from the sun. When we cannot put ^ ^ 0, our formulas
may also be applied to' the case in which the heliocentric and geocentric places
are referred, not to the ecliptic, but to any other plane whatever ; it will only be
necessary to suppress the terms longitude and latitude : moreover, account can
be immediately taken of the parallax, if only, the heliocentric place of the earth
is referred, not to the centre, but to a point on the surface. Let us put, moreover,
rcos/3=r/, J cosb = J', EcosB=:B'.
Now by referring the place of the heavenly body and of the earth in space to
three planes, of which one is the ecliptic, and the second and third have their
poles in longitude JV and JV-\- 90°, the following equations immediately present
themselves : —
/ cos {l — JV) — R' cos [L — N)= A' cos {I — N)
r' sin {l — N) — R' sin {L — N)^ A' sin {l—N)
r' tan {-i — R' tan B =z J' tan h,
in which the angle N is wholly arbitrary. The first and second equations will
determine directly I — N and z/', whence h will follow from the third ; from h
and J' you will have A. That the labor of calculation may be as convenient as
possible, we determine the arbitrary angle N in the three following ways: —
I. By putting iV= X, we shall make
and I ^
^sin(^ — Z) = P, -^cos(l — Z) — 1= ^,
72 EELATIONS PERTAINING SIMPLY" [BoOK ].
p
tan(^ — L) = -^
A' P Q
R' sin (l—L)~ cos (^ — L)
tanJ=:
-, tan fl — tan B
n. By putting N^ ?., we shall make
^•sin(2. — X) = P, l_:^cos(X — Z)=^,
and we shall have,
P
tan (^ — V)
Q
A' P Q
r' &m{l — l) cos {I— I)
tan^:
7?'
tan /3 r tan B
m. By putting iV= ^ (^ + ^)j ^ and J' will be found by means of the
equations
tan {l—^{l-{. L)) = ^^±|! tan ^{l—L)
., (/ + iy)sin^(X — Z) {r' — R')Q.o%\{}. — L)
~ sin(Z— 1(;L + X)) "~ cos(;-i(;i + Z)) '
and afterwards h, by means of the equation given above. The logarithm of the
fraction
t'^R'
r' — R'
is conveniently computed if -j is put = tan C, whence we have
:-^ = tan(45° + c:).
In this manner the method IH. for the determination of I is somewhat shorter
than I. and 11.; but, for the remaining operations, we consider the two latter
preferable to the former.
Sect. 2.]
TO POSITION IN SPACE.
73
63.
For an example, we continue further the calculation carried to the helio-
centric place in article 51. Let the heliocentric longitude of the earth,
24:° 19' 4:9" M = L, and log J? = 9.9980979, correspond to that place; we put
the latitude =0. We have, therefore, l — L = —17°24'20".Q7,logB' = B,
and thus, according to method IL,
log^ .... 9.6729813 log(l— ^) .... 9.6526258
- log sin (^ — Z) . 9.4758653^ l—Q— 0.4493925
^ logcos(X — Z) . 9.9796445 Q= 0.5506075
--logP . . . . 9.1488466 w
log^ .... 9.7408421
Hence i—X = — 14^21' 6".75 whence 1= 352°34'22".23
log^ . . . . 9.7546117 whence log z/' . . . 0.0797283
log tan/? . . . 8.8020996 w log cos ^ 9.9973144
logtanJ . . . 9.0474879?? log J 0.0824139
b = — 6°2r55".07
According to method HI., from log tan ^ = 9.6729813, we have C = 25°13'6''.31,
and thus,
log tan (45° + 0 . • . 0.4441091
logUn ^l — L) . . . 9.1848938JZ
logtan(^— U — ^Z) . 9.6290029W
l—n — iL=:— 23° 3'16".79 I whence^=:352°34'22'^225.
H+iX:= 15 37 39.015]
64.
We further add the following remarks concerning the problem of article 62.
I. By putting, in the second equation there given,
JSr=l, JSrz=zL, N=il,
10
74 RELATIONS PERTAINING SIMPLY [BoOK I.
there results
Ii'sm{X — Z) = J'sm{l—X)
r'sm{X—L) = //'sm{l—L)
rsm{l — 1}=E' sin {I — X) .
The first or the second equation can be conveniently used for the proof of the
calculation, if the method I or II. of article 62 has been employed. In our
example it is as follows : —
logsin(X — X) . . . 9.4758653 ;z /— Z = — 3r45'26".82
log4 9.7546117
9.7212536^2
logsin(?— X) . . . 9.7212536 n
n. The sun, and the two points in the plane of the ecliptic which are the
projections of the place of the heavenly body and the place of the earth form a
plane triangle, the sides of which are J', B', /, and the opposite angles, either
l—L,l—l, 180° — ^+X, or L — l, I— I, and im°'—L-\-l; from this the
relations given in I. readily follow.
III. The sun, the true place of the heavenly body in space, and the true place
of the earth will form another triangle, of which the sides will be J, R, r : if,
therefore, the angles opposite to them respectively be denoted by
S,T, IW — S—T,
we shall have
&mS sinT sin {S^ T)
A E r ''
The plane of this triangle will project a great circle on the celestial sphere, in
which will be situated the heliocentric place of the earth, the hehocentric place
of the heavenly body, and its geocentric place, and in such a manner that the
distance of the second from the first, of the third from the second, of the third
from the first, counted in the same direction, will be respectively, S,T, 8 -\- T.
IV. The following differential equations are derived from known differential
variations of the parts of a pla,ne triangle, or with equal facihty from the formu-
las of article 62: —
Sect. 2.] to position m space. 75
J, / cos 5 sin 5 sin (X — 0 Ji i r' cos'b ,^ , cos^b ., ^ /i 7\ i^ i\ -i i
d^= ^— ^ ^d^ + -,^^d/3 + -^^(tafl/3 — cos(2. — Z)taii^)dr',
in which the terms which contain dr dA' are to be multiplied by 206265, or the
rest are to be divided by 206265, if the variations of the angles are expressed in
seconds.
Y. The inverse problem, that is, the determination of the hehocentric from
the geocentric place, is entirely analogous to the problem solved above, on which
account it would be superfluous to pursue it further. For all the formulas of
article 62 answer also for that problem, if, only, all, the quantities which relate to
the heliocentric place of the body being changed for analogous ones referring to
the geocentric place, L -j- 180° and — B are substituted respectively for L and B,
or, which is the same thing, if the geocentric place of the sun is taken instead of
the heliocentric place of the earth.
65.
Although in that case where only a very few geocentric places are to be
determined from given elements, it is hardly worth while to employ all the
devices above given, by means of which we can pass directly from the eccentric
anomaly to the geocentric longitude and latitude, and so also to the right ascen-
sion and declination, because the saving of labor therefrom would be lost in
the preliminary computation of the multitude of auxiliary quantities; still, the
comljination of the reduction to the ecliptic with the computation of the geocen-
tric longitude and latitude will afford an advantage not to be despised. For if the
ecliptic itself is assumed for the plane of the coordinates s, and the poles of
the planes of the coordinates x,y, are placed in S2, 90° -[- ^, the coordinates are
very easily determined without any necessity for auxiliary quantities. We have,
a; = r cos u
y=. r cos^sin^«
s = r sin i sin u
X=E'cos{L—Q)
Y=B'sm{L—Q)
Z=i R' tan B
x — X=J'cos{l—i
z — Z=J'idJib.
76
RELATIONS PERTAINING SIMPLY
When B = (), tlien B'=R,Z^
solved as follows : —
L
[Book I.
According to these formulas our example is
213°12'0'^32.
logi^' . . .
log cos (Z — 9,)
log sin (X — Q,)
logr 0.3259877
log cos M 9.9824141 w
log sin M 9.4454714«
log a; 0.3084018 w logX 9.9207006 w
9.9980979
9.9226027?z
9.7384353W
log r sin ■
log cos /
log sin i
9.7714591^
9.9885266
9.3557570
'y
9.7599857^2
9.1272161W
log^
Hence follows
\ogX^—X) . . . 0.0795906 w
\og{y — Y) . . . 8.4807165W
whence (/— a) = 18r26'33".49
\ogJ' ...... 0.0797283
log tan 5 9.0474878 «
logZ 9.7365332re
Z= 0
352°34'22".22
— 6 21 65.06
The right ascension and declination of any point whatever in the celestial
sphere are derived from its longitude and latitude by the solution of the spherical
triangle which is formed by that point and by the north poles of the ecliptic and
equator. Let e be the obhquity of the ecliptic, / the longitude, h the latitude, a
the right ascension, d the declination, and the sides of the triangle will be e,
90° — b, 90° — d ; it wiU be proper to take for the angles opposite the second
and tMrd sides, 90° -\-a, 90° — /, (if we conceive the idea of the spherical triangle
in its utmost generahty) ; the thhd angle, opposite e, we will put= 90° — E. "We
shall have, therefore, by the formulas, article 54,
Sect. 2.] TO POSITION in space. 77
sin (45° — H) sin i (^ -f a) = sin (45° + I /) sin (45° — I (e + h)}
sin (45° — ^ d) cos i{i;-\-a) = cos (45° -f- h I) cos (45° — i{e — b))
cos (45°— hd)sm^:E—a) = cos (45° + i I) sin (45° — ^ (e — b))
cos (45°— i (J) cos ^i:—a) = sin (45° + i I) cos (45° — i (e + b))
The first two equations wUl give ^ {IJ-\- a) and sin (45° — 2 d); the last two,
^{U — a) and cos (45° — id); from i(^-f-^) ^^^ 2 (^ — a) will be had a, and,
at the same time, U; from sin (45° — ^ (5") or cos (45° — I d), the agreement of
which will serve for proving the calculation, will be determined 45° — id, and
hence d. The determination of the angles i [U -\- a), i [E — a) by means of
their tangents is not subject to ambiguity, because both the sine and cosine of the
angle 45° — id must be positive.
The differentials of the quantities a, d, from the changes of I, b, are found
according to known principles to be,
a = ^—dl ^ab
cos 0 cos 0
did ^ COS E cosb dl-\-siia.E db.
67.
Another method is required of solving the problem of the preceding article
from the equations
cos 6 sin ^ = sin e tan b -\- cos I tan a
sin d = cos £ sin 5 -|- sin t cos ^ sin /
cos ^ cos / = cos a cos (^ .
The auxiliary angle 6 is determined by the equation
, . tan S
tan^= -r-^,
and we shall have
tan«=^^^^^+^^
COSP
tan d = sin « tan (e -|- ^) ,
to which equations may be added, to test the calculation,
R cos 5 COS Z r> COS (s -\~ 6) cos b sin I
coso = -or COSO:= — / . .
cos a cos 0 sin a
78 RELATIONS PERTAINING SIMPLY [BoOK 1.
This ambiguity in the determination of a by the second equation is removed by
this consideration, that cos a and cos I must have the same sign.
This method is less expeditious, if, besides a and ^, E also is requned : the most
convenient formula for determining this angle will then be
-r, sin £ COS a sin £ cos I
cosJE:= ^— = — -.
cos 0 COS 0
But JE cannot be correctly computed by this formula when + cos U differs but
little from imity ; moreover, the ambiguity remains whether JE should be taken
between 0 and 180°, or between 180° and 360°. The inconvenience is rarely
of any importance, particularly, since extreme precision in the value of U is not
required for computing differential ratios j but the ambiguity is easily removed
by the help of the equation
cos b cos d sin ^ = cos e — sin J sin d,
which shows that U must be taken between 0 and 180°, or between 180° and
360°, according as cose is greater or less than sin i sin^ : this test is evidently not
necessary when either one of the angles h, d, does not exceed the limit 66° 32' ;
for in that case sin U is always positive. Finally, the same equation, in the case
above pointed out, can be appHed to the more exact determination of U, if it
appears worth while.
68.
The solution of the inverse problem, that is, the determination of the longi-
tude and latitude from the right ascension and declination, is based upon the same
spherical triangle ; the formulas, therefore, above given, will be adapted to this
purpose by the mere interchange of b with d, and of /with — a. It will not be
unacceptable to add these formulas also, on account of their frequent use :
According to the method of article 66, we have,
sin (45° — i h) sin h (^— 7) = cos (45° + h a) sin (45°— ^ (^ +'^))
sin (45° — h h) cos k{E—l)=. sin (45^ + I a) cos (45° — ^e — d))
cos (45° —hb) sin h{E-\-l) = sin (45° + ^ «) sin (45° — I (e — d))
co8(45°— i^)cos|(^+/)=cos(45°+ia)cos(45° — i(£+^)).
Sect. 2.] to position m space. 79
As in the other method of article 67, we will determine the auxiliary angle ^
by the equation
. y tan 5
tan ^ = -. — ,
sin a'
and we shall have
, , cos (t — s) tana
tan^= — 5^^ ^
cos Q
tan b = sin Ha,n (^ — 6).
For proving the calculation, may be added,
, cos 5 cos a cos (C — e) cos d sin a
COS b =^ J— = — ^ y^^^i •
cos I cos f sm I
For the determination of E, in the same way as in the preceding article, the fol-
lowing equations will answer : —
7-, sin £ cos a sin e cos I
COSjE'= j—= -r—
cos 0 cos 0
COS b COS ^ sin ^ := cos e — sin J sin d .
The differentials of /, h, will be given by the formulas
^ , sin^cosfi 1 I cos^ T ft
dl= -,— d 0: -^ r d o
cos 0 ' cos 0
db = — COS -E" COS ^ d « 4- sin -S" d d .
We will compute, for an example, the longitude and latitude from the right
ascension 355° 43' 45".30=r a, the declination — 8° 47' 25" = d^, and the obliquity
of the echptic 23° 27' 59".26 = e. We have, therefore, 45^ + i a = 222° 51' 52".65,
45° — i(e+(^) = 37°39'42".87, 45° — I (e — d") = 28°52'17".87; hence also,
.ogcos(45°+2ia) . . 9.8650820W log sin (45° + i a) . . 9.8326803?2
.ogsin(45° — |(e + d)) 9.7860418 logsin (45°— i (e — (^)) 9.6838112
.ogcos(45°— i(e + ^)) 9.8985222 logcos(45° — ^e — (^)) 9.9423572
.ogsin(45° — i^)sin^(^— ;) . . 9.6 511238 w
ogsin(45° — f^)cos|(^— /) . . 9.7750375 w ■
whence ^ (^— /) = 216''56'5".39 ; log sin (45° —^b) = 9.8723171
80
RELATIONS PERTAIOTNG SIMPLY
[Book L
logcos(45°— i5)sm^^ + Z) . . 9.5164915 w
logcos(45°— ^J)cos^(^+/) . . 9.7636042 w
whence h{E^l)=^ 209° 30'49".94 : log cos (45° —hh) = 9.8239669.
Therefore, we have E = 426° 26' 55".33, l = — T 25' 15".45, or, what amounts
to the same thing, ^ == 66°26'55".33, /= 352°34'44".55; the angle 45° — i ^,
obtained from the logarithm of the sine, is 48°10'58".12, from the logarithm of
-the cosine, 48°10'58".17, from the tangent, the logarithm of which is their differ-
ence, 48° 10' 68".14 ; hence ^ === — 6° 21' 56".28.
According to the other method, the calculation is as follows : —
log tan d
9.1893062W
log tan C .... 0.3173270
C. log cos ^
log cos {t,—
log tan a
0.3626190
9.8789703
8.8731869«
64°ir6".83
40 49 7 .57
log tan? 9.1147762 w
1= 352°34'44".50
log sin/ 9.1111232 w
logtan(C — e) . . 9.9363874
log tan ^ 9.0475106 ?z
h= — 6°21'56".26.
For determining the angle E we have the double calculation
log sine 6.6001144
log cos/ 9.9963470
Clog cos ^ .... 0.0051313
log cos ^ 9.6015927
log sin e . . .
. 9.6001144
log cos a . . .
, 9.9987924
C. log cos J . .
. 0.0026859
log cos E
whence E =
. 9.6015927
66°26'55".35.
70.
Something is stiU to be added concerning the paralhx and aberration, that
nothing requisite for the computation of geocentric places maj be wanting.
We have already described, above, a method, according to which, the place
affected by parallax, that is, corresponding to any point on the surface of the
Sect. 2.] to position m space. 81
earth, can be determined directly with the greatest facility ; but as in the com-
mon method, given in article 62 and the following articles, the geocentric place is
commonly referred to the centre of the earth, in which case it is said to be free
from parallax, it will be necessary to add a particular method for determining the
parallax, which is the difference between the two places.
Let the geocentric longitude and latitude of the heavenly body with reference
to the centre of the earth be I, fi ; the same with respect to any point whatever
on the surface of the earth be I, h ; the distance of the body from the centre of
the earth, r ; from the point on the surface, //; lastly, let the longitude L, and the
latitude B, correspond to the zenith of this point in the celestial sphere, and let
the radius of the earth be denoted by R. Now it is evident that all the equations
of article 62 will be applicable to this place also, but they can be materially
abridged, since in this place R expresses a quantity which nearly vanishes in
comparison with r and J. The same equations evidently will hold good if 1,1, L
denote right ascensions instead of longitudes, and /?, h, B, declinations instead of
latitudes. In this case / — 1, h — §, will be the parallaxes in right ascension and
declination, but in the other, parallaxes in longitude and latitude. If, accord-
ingly, R is regarded as a quantity of the first order, I — l,h — (i, J — r, will be
quantities of the same order; and the higher orders being neglected, from the
formulas of article 62 will be readily derived : —
J , , R cos B sin (X — L)
r cos ^
n. h-§=^-^^^-^{i^n(ioo^{l-L)-i^nB)
m. J — r = — i? COS B sin /5 ( cotan j3 cos (X — L) -\- tan Bj .
The auxiliary angle ^ being so taken that
tan^ *^"^
cos {X — Ly
the equations 11, and III. assume the following form : —
-rj 7 o R COS B COS {I — L) sin ((3 — d) R sm B bvo. {B — d)
^ r cos (9 r&md
j-pr J i2cos-Bcos(^ — L)cos(§ — 6) ^sinjBcos(^ — d)
cos (? sin 5 '
11
82
RELATIONS PERTAINING SIMPLY
[Book 1.
Further, it is evident, that in I. and II., in order that / — X and b — (i may be
had in seconds, for B, must be taken the mean parallax of the sun in seconds ;
but in in., for B, must be taken the same parallax divided by 206265". Finally,
when it is required to determine in the inverse problem, the place free from
parallax from the place affected by it, it will be admissible to use J, I, b, instead
of r, I, fi, in the values of the parallaxes, without loss of precision.
Example. — Let the right ascension of the sun for the centre of the earth
be 220°46'44".65 = X,the declination,— 15° 49'43".94=^, the distance, 0.9904311
=^ r : and the sidereal time at any point on the surface of the earth expressed
in degrees, 78° 20' 38" = i:, the elevation of the pole of the point, 45°2r57" = ^,
the mean solar parallax, 8".6 = R. The place of the sun as seen from this point,
and its distance from the same, are required.
logR 0.93450
logcos^ 9.84593
C.logr 0.00418
Clog cos ^i .... 0.01679
logsin(^ — Z) . . . 9.78508
loga— ^) . . . . 0.58648
1—1= 4-3".86
1= 220°46'48".51
log tan 5 0.00706
logcos(X — X) . . . 9.89909W
log tan ^ 0.10797m
&= 127° Sr 0''
(i — &= —143 46 44
logR 0.93450
log sin ^ 9.85299
C.logr . . . . .' . 0.00418
Clog sin ^ 0.10317
logsin(^ — ^) . . . 9.77152 w
\og{b — ii) .... 0.66636 ^^
b — i^= — 4".64
b= — 15°49'48".58
log(^_/i}) .... 0.66636 «
log cot (/i — ^) . . . 0.13522
logr 9.99582
logl" 4.68557
log(r — z/) . . . . 5.48297"^
r — A= —0.0000304
J= 0.9904615
71.
Tlie aberration of the fixed stars, and also that part of the aberration of com-
ets and planets due to the motion of the earth alone, arises from the fact, that
the telescope is carried along with the earth, whUe the ray of light is passing
Sect. 2.] to position m space. 83
along its optical axis. The observed place of a heavenly body (which is called
the apparent, or affected by aberration), is determined by the direction of the
optical axis of the telescope set in such a way, that a ray of light proceeding
from the body on its path may impinge upon both extremities of its axis : but this
direction differs from the true direction of the ray of light in space. Let us con-
sider two moments of time t, t', when the ray of light touches the anterior ex-
tremity (the centre of the object-glass), and the posterior (the focus of the object-
glass) ; let the position of these extremities in space be for the first moment a, I ;
for the last moment a',b'. Then it is evident that the straight line ab' is the true
direction of the ray in space, but that the straight line ab or a'b' (which may be
regarded as parallel) corresponds to the apparent place : it is perceived without
dif&culty that the apparent place does not depend upon the length of the tube.
The difference in direction of the right lines b'a, ba, is the aberration such as exists
for the fixed stars : we shall pass over the mode of calculating it, as well known.
This difference is still not the entire aberration for the wandering stars : the
planet, for example, whilst the ray which left it is reaching the earth, itself
changes its place, on which account, the direction of this ray does not correspond
to the true geocentric place at the time of observation. Let us suppose the ray
of light which impinges upon the tube at the time t to have left the planet at the
time T ; and let the position of the planet in space at the time T be denoted by
P, and at the time thj p ; lastly, let A be the place of the anterior extremity of
the axis of the tube at the time T. Then it is evident that, —
1st. The right line AP shows the true place of the planet at the time T',
2d. The right line ap the true place at the time t ;
3d. The right line ba or b'a' the apparent place at the time t or f (the differ-
ence of which may be regarded as an infinitely small quantity) ;
4th. The right line b'a the same apparent place freed from the aberration of
the fixed stars.
Now the points P, a, b', lie in a straight line, and the parts Pa, ab', will be
proportional to the intervals of time t — T,f — t, if light moves with an uni-
form velocity. The interval of time f — T is always very small on account of
the immense velocity of light j within it, it is allowable to consider the motion
84 RELATIONS PERTAINING SMPLY [BoOK I.
of the earth as rectilinear and its velocity as uniform : so also A, a, a will lie in a
straight line, and the parts Aa, ad will likewise be proportional to the intervals
t — T, t' — t. Hence it is readily inferred, that the right lines AP, h'a' are paral-
lel, and therefore that the first and third places are identical.
The time t — T, within which the light traverses the mean distance of the
earth from the sun which we take for unity, will be the product of the distance
Pa into 493^ In this calculation it will be proper to take, instead of the dis-
tance Pa, either PA or pa, since the difference can be of no importance.
From these principles follow three methods of determining the apparent place
of a planet or comet for any time t, of which sometimes one and sometimes
another may be preferred.
I. The time in which the light is passing from the planet to the earth may be
subtracted from the given time ; thus we shall have the reduced time T, for which
the true place, computed in the usual way, wiQ be identical with the apparent
place for t For computing the reduction of the time t — T,\i is requisite to
know the distance from the earth ; generally, convenient helps will not be want-
ing for this purpose, as, for example, an ephemeris hastily calculated, otherwise it
will be sufficient to determine, by a preliminary calculation, the true distance for
the time t in the usual manner, avoiding an unnecessary degree of precision.
II. The true place and distance may be computed for the instant t, and,
from this, the reduction of the time t — T, and hence, with the help of the daily
motion (in longitude and latitude, or in right ascension and declination), the re-
duction of the true place to the time T.
III. The heliocentric place of the earth may be computed for the time /; and
the heliocentric place of the planet for the time T : then, from the combination
of these in the usual way, the geocentric place of the planet, which, increased
by the aberration of the fixed stars (to be obtained by a well-known method, or
to be taken from the tables), will furnish the apparent place sought.
The second method, which is commonly used, is preferable to the others,
because there is no need of a double calculation for determining the distance,
but it labors under this inconvenience, that it cannot be used except several
places near each other are calcalatcd, or are known from observation ; otherwise
it would not be admissible to consider the diurnal motion as o-iven.
Sect. 2.] to position in space. 85
The disadvantage with which, the first and third methods are incumbered, is
evidently removed when several places near each other are to be computed.
For, as soon as the distances are known for some, the distances next following
may be deduced very conveniently and with sufficient accuracy by means of
familiar methods. K the distance is known, the first method will be generally
preferable to the third, because it does not require the aberration of the fixed
stars ; but if the double calculation is to be resorted to, the third is recommended
by this, that the place of the earth, at least, is retained in the second calculation.
What is wanted for the inverse problem, that is, when the true is to be derived
from the apparent place, readily suggests itself According to method I., you will
retain the place itself unchanged, but will convert the time t, to which the given
place corresponds as the apparent place, into the reduced time T, to which the
same will correspond as the true place. According to method II., you, will retain
the time t, but you will add to the given place the motion in the time t — iT, as
you would wish to reduce it to the time t-^{t — T). According to the method
ni., you will regard the given place, free from the aberration of the fixed stars,
as the true place for the time T, but the true place of the earth, answering to
the time t, is to be retained as if it also belonged to T. The utihty of the third
method will more clearly appear in the second book.
Finally, that nothing may be wanting, we observe that the place of the sun is
affected in the same manner by aberration, as the place of a planet : but since
both the distance from the earth and the diurnal motion are nearly constant, the
aberration itself has an almost constant value equal to the mean motion of
the sun in 493', and so = 20".25; which quantity is to be subtracted from the
true to obtain the mean longitude. The exact value of the aberration is in the
compound ratio of the distance and the diurnal motion, or what amounts to the
same thing, in the inverse ratio of the distance ; whence, the mean value must be
diminished in apogee by 0".34, and increased by the same amount in perigee.
Our solar tables already include the constant aberration — 20".25 ; on which
account, it will be necessary to add 20".25 to the tabular longitude to obtain the
true.
86 RELATIONS PERTAINING SBIPLY [BoOK 1.
72.
Certain problems, which are in frequent use in the determination of the orbits
of planets and comets, will bring this section to a close. And first, we will revert
to the parallax, from which, in article 70, we showed how to free the observed
place. Such a reduction to the centre of the earth, since it supposes the distance
of the planet from the earth to be at least approximately known, cannot be made
when the orbit of the planet is wholly unknown. But, even in this case, it is pos-
sible to reach the object on account of which the reduction to the centre of the
earth is made, since several formulas acquire greater simplicity and neatness
from this centre lying, or being supposed to lie, in the plane of the ecliptic,
than they would have if the observation should be referred to a point out of the
plane of the ecliptic. In this regard, it is of no importance whether the obser-
vation be reduced to the centre of the earth, or to any other point in the plane
of the ecliptic. Now it is apparent, that if the point of intersection of the
plane of the ecliptic with a straight line drawn from the planet through the true
place of observation be chosen, the observation requires no reduction whatever,
since the planet may be seen in the same way from all points of this line:* where-
fore, it will be admissible to substitute this point as a fictitious place of observa-
tion instead of the true place. We determine the situation of this point in the
following manner : —
Let X be the longitude of the heavenly body, /5 the latitude, J the distance,
all referred to the true place of observation on the surface of the earth, to
the zenith of which corresponds the longitude /, and the latitude b ; let, more-
over,-7t be the semidiameter of the earth, L the heliocentric longitude of the cen-
tre of the earth, B its latitude, B its distance from the sun ; lastly, let X' be the
hehocentric longitude of the fictitious place, li' its distance from the sun, J -\-d
* If the nicest accuracy should be wanted, it would be necessary to add to or subtract from the given
time, the interval of time in which light passes from the true place of observation to the fictitious, or from
the latter to the former, if we are treating of places affected by aberMion : but this difference can
scarcely be of any importance unless the latitude should be very small.
Sect. 2.] to position in space. 87
its distance from the heavenly Ibody. Then, TV denoting an arbitrary angle, the
following equations are obtained withaut any difficulty : —
R' cos {L'—N) -f d cos (i cos {I —N) =EcosB cos {L—N) + tt cos 3 cos {l—N)
R' sin {L'—N) + d cos {i sin {I —N) = R cos B sin {L—N) -f tt cos ^ sin {l—N)
d mi ^ =^ R sin B -\- n mil .
Putting, therefore,
L (i? sin ^ -|- ^ ^i'^ ^) cotan ^ =fi,
we shall have
n. R' cos {L'—N) = R cos ^ cos {L —N)-\- n cos h cos {l—N)—fi cos (^ — iV)
m. ^' sin {L' —N) = RcosB sin (Z —N) + tt cos ^ sin {l—N) —^ sin {I —N)
IV. (^ = -^,.
cos p
From equations 11. and IH., can be determined R' and L', from IV., the inter-
val of time to be added to the time of observation, which in seconds will be
= 493 d.
These equations are exact and general, and will be applicable therefore when,
the plane of the equator being substituted for the plane of the ecliptic, L, L', I, X,
denote right ascensions, and B, h, (i declinations. But in the case which we are
specially treating, that is, when the fictitious place must be situated in the eclip-
tic, the smallness of the quantities B, it, LI — L, still allows some abbreviation of
the preceding formulas. The mean solar parallax may be taken for n ; B, for
sin B ; 1, for cos B, and also for cos {L' — L)-, L' — L, for sin {L' — L). In this
way, making N=^ L, the preceding formulas assume the following form : —
I. jU, = {RB -\- n sin h) cotan /3
n. ^' = 7? -|- TT cos ^ cos (/ — L) — lit cos (X — L)
j-rj jf J 7rcos5sin(Z — X) — jit sin (i — X)
R'
Here B, n, L' — L are, properly, to be expressed in parts of the radius ; but it is
evident, that if those angles are expressed in seconds, the equations I., III. can be
retained without alteration, but for 11. must be substituted
TV jy j^ 7t COS b COS {I — L) — n cos Q. — L)
IC—M-\ 206265" *
RELATIONS PERTAINING SBIPLY
[Book I.
Lastly, in the formula HI., R may always be used in place of the denominator i?'
without sensible error. The reduction of the time, the angles being expressed
in seconds, becomes
493'. fi
206265". cos /3'
log; 7T . . . .
. . 0.93450
log sin 5 . . .
. . 9.86330
log n smb . .
. . 0.79780
73.
Umm2)le. — Let X r= 354° 44' 54", /5 = — 4° 59' 32", ^=24° 29', ^ =: 46° 63',
L'= 12° 28' 54", B = ^ 0".49, B = 0.9988839, n = 8".60. The calculation is as
follows : —
logE 9.99951
log^ 9.69020
\ogBE 9.68971
Hencelog(^i?-l-7rsinJ) . 0.83040
logcotan^ .... 1.05873 w
logjiA 1.88913?^
logTT 0.93450
log cos 5 9.83473
logl" 4.68557
log cos (/—X) . . . 9.99040
5.44520
number + 0.0000279
Hence is obtained B' = B-{- 0.0003856 = 0.9992695. Moreover, we have
log|i* 1.88913 jz
logr 4.68557
logcos(X — Z) . . . 9.97886
6.55356?i
number — 0.0003577
logTTcos^ 0.76923
logsin(/ — X) . . . 9.31794
C.logi^' 0.00032
0.08749
number 4- r.22
log lit 1.88913 w
logsin(X — Z) . . 9.48371W
C.logi?' 0.00032
1.37316
number + 23".61
Sect. 2.] to position in space. 89
Whence is obtained L'=^L — 22".39. Finally we have
log;i* 1.88913W
Clog 206265 .... 4.68557
log 493 2.69285
Clog cos/? 0.00165
9.26 920 w,
whence the reduction of time = — 0M86, and thus is of no importance.
74.
The other problem, to deduce the heliocentric place of a heavenly lody in its orlit
from the geoceiitric place and the sanation of the plane of the orhit, is thus far similar to
the precediugj that it also depends upon the intersection of a right line drawn
between the earth and the heavenly body with the plane given in position. The
solution is most conveniently obtained from the formulas of article 65, where the
meaning of the sjnnbols was as follows : —
L the longitude of the earth, R the distance from the sun, the latitude B we
put =0, — since the case in which it is not =: 0, can easily be reduced to this by
article 72, — whence R' = R, I the geocentric longitude of the heavenly body, b
the latitude, /I the distance from the earth, r the distance from the sun, u the
argument of the latitude, Q, the longitude of the ascending node, i the inclination
of the orbit. Thus we have the equations
I. rcosM — ^cos(X — 9>)=zJ coBh c,ob{1 — 9,)
n. r cos z sin ?{ — i? sin (Z — Q ) = // cos b sin [I — 9, )
ni. r sin i sin ^f = z/ sin J .
Multiplying equation I. by sin (Z — Q>) sin b, 11. by — cos (Z — -9,) sin b, IH. by
— sin (Z — I) cos b, and adding together the products, we have
cos u sin [L — Q, ) sin J — sin u cos i cos (X — ^ ) sin J — sin if sin «' sin [L — I) cos J = 0,
whence
lY. i^nu= sin(L-Q)smh
cos i cos (X — ^ ) sin 5 -[- sin i sin (L — ^) cos i
12
90 RELATIONS PERTAINING SIMPLY [BoOK I.
Multiplj-ing likewise I. by sin [l — 9,), 11. by — cos [l — Q), and adding together
the products, we have
^j- -ffi sin (L — I)
sin u cos i cos (I — Q,) — cos m sin (/ — Q)'
The ambiguity in the determination of u by means of equation IV., is removed
by equation HI., which shows that it, is to be taken between 0 and 180°, or be-
tween 180° and 360° according as the latitude b may be positive or negative ;
but if J = 0, equation V. teaches us that we must put ic = 180°, or u = 0, accord-
ing as sin [L — I) and sin {I — Q) have the same or different signs.
The numerical computation of the formulas IV. and V. may be abbreviated in
various ways by the introduction of auxiliary angles. For example, putting
tan5c^(Z-^)^^^^^
sin (Z — I) '
we have
sin {A~\-i) '
putting
tan i sin (i — I) , -r^
.j.^ - / = tan B,
cos{L—Q) '
we have
sin (iJ-j-o) cos I
In the same manner the equation V. obtains a neater form by the introduction
of the angle, the tangent of which is equal to
COS z tan u, or — ^ — : — -.
' cos^
Just as we have obtained formula V. by the combination of I., 11., so by a combina-
tion of the equations IT., LI., we arrive at the following : —
^^ igsin(Z-a) .
sin u (cos i — sin i sin (l — £1) cotan b) '
and in the same manner, by the combination of equations I., IH., at this ;
i?cos(Z— g2)
sm u sill t cos
(^—S^) cotan J'
Sect. 2.] to position in space. 91
both of whicli, in the same manner as Y., may be rendered more simple by the
introduction of auxiliary angles. The solutions resulting from the preceding
equations are met with in Von Zach Monatliche Correspondens, Vol. V. p. 540, col-
lected and illustrated by an example, wherefore we dispense with their further
development in this place. If, besides u and r, the distance J is also wanted, it
can be determined by means of equation III.
75.
Another solution of the preceding problem rests upon the truth asserted in arti-
cle 64, in., — that the heHocentric place of the earth, the geocentric place of the
heavenly body and its heliocentric place are situated in one and the same great
circle of the sphere. In fig. 3 let these places be respectively T, G, H; further,
let 9> be the place of the ascending node ; 9>T, 9,11, parts of the ecliptic and
orbit ; GP the perpendicular let fall upon the ecliptic from G, which, therefore,
wiUbe=^. Hence, and from the arc PT=i/ — ^will be determined the angle T
and the arc TG. Then in the spherical triangle 9, HT are given the angle 9> = i,
the angle T, and the side 9>T=^L — 9>, whence will be got the two remaining
9, R= u and TK Finally we have HG = TG— TH, and
R sin TG . R sin TH
&mHG ' smHG '
76.
In article 52 we have shown how to express the differentials of the heliocen-
tric longitude and latitude, and of the curtate distance for changes in the argu-
ment of the latitude u, the inchnation i, and the radius vector r, and subsequently
(article 64, IV.) Ave have deduced from these the variations of the geocentric
longitude and latitude, I and b : therefore, by a combination of these formulas, d I
and d^ will be had expressed by means of d^f, d^', dS2, dr. But it wiU be worth
while to show, how, in this calculation, the reduction of the heliocentric place
to the ecliptic, may be omitted in the same way as in article 65 we have
deduced the geocentric place immediately from the heliocentric place in orbit.
That the formulas may become more simple, we will neglect the latitude of
92 RELATIONS PERTAINING SIMPLY [BoOK I.
the earth, which of course can have no sensible effect in differ'ential formulas.
The following formulas accordingly are at hand, in which, for the sake of brevity,
we write w instead of/ — Q,, and also, as above, J' in the place of z^ cos J.
J' cos (0 = r cos u — M cos (X — S^ ) z= ^
J' sinb) =:^rcosismu — i?sin(Z — Q,)-=fj
A' tan 3 = r sin i sin u^=lL,\
from the diflferentiation of which result
cos (o.dJ' — z/'sin w.dto = d^
sin to . d ^' -j- ^' cos to . d w = d?^
tan5.dz/'4-^,d5==dC.
' cos 0
Hence by elimination,
— sin w . d J -1- cos oi.dti
0)^ '-^
A
■, 7 — COS to. sin 5. d 5 — sin wsin J.dj/-)- cosS.d^
A
K in these formulas, instead of ^, r], t, their values are substituted, dto
and db will appear represented by dr, du, di, dQ, ; after this, on account of
d/=dto-|-dS, the partial differentials of I and b will be as follows : —
<dZ>
sm to cos II, -j- cos to sm u cos i
il. — I ^^ I = sm to sm ?f + cos o) cos u cos ;
TTT ^'/<iZ\ . . .
ill. — ( — ■ ) = — COS 0) sm u sm i
^- (^)=l + |cos(X-g2-to) = l+|cos(Z-0
Y. J ( — j = — cos to cos u sin b — sin to sin ?< cos i sin b -\- sin u sin ^ cos b
VI. —(—) ^ cos to sin u sin h — sin to cos u cos ^' sin J 4- cos m sin i cos b
r \au/ '
Vn. — (t-.) = sin (i) sin ic sin i sin b -j- sin u cos i cos b
Vm. ;»-(^ q) = sin 5 sin (Z — Q — to) := sin J sin (L — I).
Sect. 2.] to position in space. 93
The formulas IV. and "VILL. already appear in tlie most convenient form for cal-
culation ; but the formulas L, HL, Y., are reduced to a more elegant form by
obvious substitutions, as
m.* (^) =— cos w tan h
v.* (j-) = — -^^^^{^ — ^sinJ = — — ,cos(Z — ^)sinJcosJ.
Finally, the remaining formulas 11., VI., VII., are changed into a more simple form
by the introduction of certain auxiliary angles : which may be most conveniently
done in the following manner. The auxiliary angles M, N, may be determined
by means of the formulas
tan Jf = — % , tan iV= sin to tan i = tan Jf cos to sia ^.
COSi '
Then at the same time we have
cos^M 1 4- tan^ N cos^ i 4- sin^ to sin^ i «
cos2J\r — l+tan^if ~ 'cos^i + tan^w — ^"^ »" •
now, since the doubt remaining in the determination oi M, N,hj their tangents,
may be settled at pleasure, it is evident that this can be done so that we may
have
cos M 1
^ = H- cos to,
cos iv ' '
and thence
sa\N ...
-T-Tj= = + smz.
sin if '
These steps being taken, the formulas IE., YI., VII., are transformed into the fol-
lowing : —
TT* (^^\ »* sin ft) COS (iHf — m)
^' Wu) ~ ^' sin if
VI.* (i^) = ^(j^o^ w sin 2 cos (IT— m) cos (iV — h)-\-mi{M — u) sin {N — h))
VTT * (^ ^\ ^ ®^^ " ^''^ * *^°^ ("^ — ^)
\d 1/ ^ cos i\r
94 RELATIONS PERTAINING SIMPLY [BoOK I.
These transfonnations, so far as the formulas 11. and VH. are concerned, will detain
no one, but in respect to formula VI., some explanation will not be superfluous.
From the substitution, in the first place, of M — ( Jf — u) for u, in formula YI.,
there results
— (^j = cos [M — u) (cos (a sin Jlf sin h — sin w cos/cos Jf sin h -\- sin^ cos J/cos h)
— sin [M — u) (cos to cos Jf sin5 -[- sin m cos i sin J/ sin h — sin i siniJ/cos h).
Now we have
cos w sin M=z cos^ i cos w sin M-\- sin^ i cos w sin M
= sin CO cos i cos M-\- sin^ i cos w sin M;
whence the former part of that expression is transformed into
sin i cos {M — u) (sin i cos la sin iHf sin b -\- cos Jlf cos h)
= sin i cos {M — u) (cos w sin iV^sin l)-{-coBOi cos iVcos 5)
= cos oi sin 2 cos [M — u) cos (iV — h) .
Likewise,
cos JV^ cos^ CO cos iV-j- sin^ co cos 1^=^ cos co cos 31 -\- sin co cos i sin iHf ;
whence the latter part of the expression is transformed into
— sin {M — u) (cos iV^sin b — sin iV^cos b) = sin {31 — it) sin (iV — b) .
The expression YI.* follows directly from this.
The auxiliary angle 3f can also be used in the transformation of formula I.,
which, by the introduction of 31, assumes the form
\dr/ A' sin M
from the comparison of which with formula I * is derived
— E sin [L — I) sin 3f= r sin co sin ( Jf — u) ;
hence also a somewhat more simple form may be given to formula 11.*, that is,
That formula VI.* may be still further abridged, it is necessary'- to introduce
a new auxiliary angle, which can be done in two ways, that is, either by putting
Sect. 2.] TO position in space. 95
, 7-, tan (M—u) , ^ tan (N'—h)
tan P = — ^ — ^-^, or tan Q = — ^^ — ^^ ;
cos CO sin I ' ^ cos to sin i '
from which results
VT * * {—\ ^ ^^^ {M— u) cos {N— h — P) r sin {N— b) cos {M— u — Q)
Vdw/ ^sinP ^sin^
The auxiliary angles M,N,P, Q, are, moreover, not merely fictitious, and it would
be easy to designate what may correspond to each one of them in the celestial
sphere ; several of the preceding equations might even be exhibited in a more
elegant form by means of arcs and angles on the sphere, on which we are less
inclined to dwell in this place, because they are not sufficient to render superflu-
ous, in numerical calculation, the formulas above given.
77.
What has been developed in the preceding article, together with what we
have given in articles 15, 16, 20, 27, 28, for the several kinds of conic sections,
wiU furnish aU which is required for the computation of the dififerential varia-
tions in the geocentric place caused by variations in the individual elements.
For the better illustration of these precepts, we will resume the example treated
above in articles 13, 14, 51, 63, 65. And first we will express dl and d^ in terms
of 6.r, du, dz, do,, according to the method of the preceding article; which cal-
culation is as foUows : —
log tan 0) . 8.40113 log sin w . 8.40099 ?z logtsin (M—u) 9.41932 re
log cos 2 . 9.98853 log tan ^ . 9.36723 log cos to sin 2 . 9.35562 w
log tan If. 8.41260 logtaniV^ . 7.76822re log tan P . . 0.06370
M = r28'52'' iV^=179°39'50" P= 49° 11' 13"
M—u=lQ^17 8 JV—b=lSQ 145 JSr—l — P= 136 50 32
96
RELATIONS PERTAINING SIMPLY
[Book I.
I*
log sm(Z-^) 9.72125
logi? . . 9.99810
Clog J' . 9.92027
(*) . .
C. log r
9.63962
9.67401
log(|-^) . 9.31363
IV.
log^ . . 9.91837
log cos{Z—I) 9.92956
n.**
(*)... 9.63962
log cot(Jf—M) 0.58068^2
m.*
M^
0.22030
. 9.84793
vn.*
log rsin^tcos^ 9.75999m
log cos(iV— J) 9.99759 «
C.log^. . 9.91759
C.loa-cosiV 0.00001 w
Y*
(**)... 9.84793
log sin 3 cos J 9.04212m
C.logr . . 9.67401
H(^!)
.56406
vm.
(*)... 9.63962
log sin 5 cos J 9.04212 m
M!^)
8.68174m
log cos (0
log tan h
9.99986m
9.04749m
lo=K^)
9.04735w
VI.**
log^ ... 0.24357
log sin (iif—w) 9.40484
log cos(i\^— ^P) 9.86301m
Clog sin P . 0.12099
M^
9.63241m
0.11152 d^-f 1.70458 d 52
0.47335 de — 0.04805 dS2.
log(l-^) . 9.67518m
These values collected give
d/=: + 0.20589 dr + 1.66073 du
d^ = -f 0.03665 dr — 0.42895 d« •
It will hardly be necessary to repeat here what we have often observed, namely,
tbat either the variations d/, d^, dii, di, dQ, are to be expressed in parts of the
radius, or the coefficients of dr are to be multiphed by 206265", if the former are
supposed to be expressed in seconds.
Denoting now the longitude of the perihelion (which in our example is
Sect. 2.] to position in space. 97
52°18'9''.30) by i7, and the true anomaly by v, the longitude in orbit will be
?f -j- ^ = t' -|- 77", and therefore du =dv-{-dIT — dQ, which value being sub-
stituted in the preceding formulas, dl and db will be expressed in terms of dr,
dv, dIT, dQ, d^. Nothing, therefore, now remains, except to express dr and dv, ac-
cording to the method of articles 15, 16, by means of the differential variations
of the elliptic elements.*
We had in our example, article 14,
log^ = 9.90355^ log©
log a 0.42244
log— ..... 0.19290
log cosy .... 9.98652
log tan 9 .... 9.40320
log sin y .... 9.84931 w
log(^) .... 0.17942 T^TT
^^^^^ log(f4). . . . 9.67495W
log a 0.42244
log cos 9 .... 9.98652
log cosy .... 9.84966
0.25862^2
2 — ecos^= 1.80085
ee= 0.06018
1.74067
log 0.24072
log:-^ 0.19290 log^-^)
log sin ^ . . . . 9.76634 ?2
log(^) .... 0.19996 w
Hence is collected
dy = + 1.51154 dif — 1.58475 dcp
dr = — 0.47310 d M— 1.81393 dg) -f- 0.80085 da ;
which values being substituted in the preceding formulas, give
d/= -f 2.41287 dM— 3.00531 dy + 0.16488 da -\- 1.66073 diZ
— 0.11152 d^ + 0.04385 dg?
dJ = — 0.66572 dilf + 0.61331 dcp-}- 0.02925 da — 0.42895 diT
— 0.47335 d»-f 0.38090 dQ.
* It will be perceived, at once, that the symbol M, in the following calculation, no longer expresses
our auxiliary angle, but (as in section 1) the mean anomaly.
18
98 RELATIONS PERTAINING SIMPLY [BoOK I
If the time, to which the computed place corresponds, is supposed to be
distant n days from the epoch, and the mean longitude for the epoch is
denoted by iV^ the daily motion by t, we shall have Jf = N-\-ni: — 77, and thus
d Jf = diV-f" ^^^"^ — ^^- ^'^ our example, the time answering to the computed
place is October 17.41507 days, of the year 1804, at the meridian of Paris : if,
accordingly, the beginning of the year 1805 is taken for the epoch, then
n=^ — 74.58493; the mean longitude for that epoch was 41°52'21".61, and the
diurnal motion, 824''.7988. Substituting now in the place of dJtf its value in
the formulas just found, the differential changes of the geocentric place, expressed
by means of the changes of the elements alone, are as follows: —
&l=z 2.41287 ^N— 179.96 dr — 0.75214 d77— 3.00531 dg) -f 0.16488 da
— 0.11152 d^ + 0.04385 da,
d^ = _ 0.66572 di\r-f 49.65 dr + 0.23677 d77-f 0.61331 d^ + 0.02935 da
_ 0.47335 d^ + 0.38090 d 9, .
K the mass of the heavenly body is either neglected, or is regarded as
known, r and a will be dependent upon each other, and so either dr or da may
be eliminated from our formulas. Thus, since by article 6 we have
we have also
dr g da
in which formula, if d-r is to be expressed in parts of the radius, it will be neces-
sary to express r in the same manner. Thus in our example we have
logT- 2.91635
logr 4.68557
log I 0.17609
Clog a .... 9.57756
log^ 7.35557W,
or, dT = — 0.0022676 da, and da = — 440.99 d-r, which value being substituted
in our formulas, the final form at length becomes : —
Sect. 2.] to position in space. 99
. d^= 2.41287 dJV— 252.67 dT — 0.75214 d77— 3.00531 dcp
— 0.11152 d/+ 0.04385 dg^,
d3 = — 0.66572 di\^+ 36.71 dT + 0.23677 di7+ 0.61331 dtp
— 0.47335 d^■+ 0.38090 dS.
In the development of these formulas we have supposed all the differentials dl,
db, dJV, dr, dIT, dcp, di, do, to be expressed in parts of the radius, but, mani-
festly, by reason of the homogeneity of all the parts, the same formulas will
answer, if all those differentials are expressed in seconds.
THIRD SECTION.
RELATIONS BETWEEN SEVERAL PLACES IN ORBIT.
78.
The discussion of the relations of two or more places of a heavenly body in
its orbit as well as in space, furnishes an abundance of elegant propositions, such
as might easily fill an entire volume. But our plan does not extend so far as to
exhaust this fruitful subject, but chiefly so far as to supply abundant facilities for
the solution of the great problem of the determination of unknown orbits from
observations : wherefore, neglecting whatever might be too remote from our pur-
pose, we will the more carefully develop every thing that can in any manner
conduce to it. We will preface these inquiries with some trigonometrical propo-
sitions, to which, since they are more commonly used, it is necessary more fre-
quently to recur.
I. Denoting by A, B, 0, any angles whatever, we have
sm ^ sin ( C— ^) -[- sin ^ sin (^ — C') 4- sin C'sin (^ — ^) = 0
cos^sin(C— ^) + cos^sm(iL — 6') + cosCsin(^ — ^) = 0.
n. If two quantities p, P, are to be determined by equations such as
p sin [A — P)=za
psm{B — P) = b,
it may generally be done by means of the formulas
p sin {B — A) sin {IT— P) = h sin {H— A)~a sin {H— B)
p sin [B — A) cos {R— P) = b cos (//— A) — a cos [H— B),
in which H is an arbitrary angle. Hence are derived (article 14, 11.) the angle
H — P, and p sin {B — A); and hence P and p. The condition added is gen-
(100)
Sect. 3.] relations betwee?^ several places in orbit. 101
erallj that p must be a positive quantity, whence the ambiguity in the deter-
mination of the angle II — -Pby means of its tangent is decided; but without
that condition, the ambiguity may be decided at pleasure. In order that the
calculation may be as convenient as possible, it will be expedient to put the arbi-
trary angle H either •= Aox =^B or :=: ^ ( J. -j- B\ In the first case the equa-
tions for determining P andj» will be
jo sin {A — P) =a,
In the second case the equations wUl be altogether analogous ; but in the third
case,
And thus if the auxiliary angle ^ is introduced, the tangent of which z=-, P wiU
be found by the formula
tan {hA^hB — P) = tan (45° + C) tan h {B—A),
and afterwards p by some one of the preceding formulas, in which
I (^ _ a) = COS (45° + 1) v/^, = ^^^^i^;±^ - ^^^^^^^
^ ^ ^ I -''V sin 2; sm4y/2 cos C V^
in. If p and P are to be determined from the equations
jt?cos(^ — P) ■=^a,
p(ios{B — P)=h,
every thing said in 11. could be immediately applied provided, only, 90° -|- A
W -\-B were written there throughout instead of A and B : that their use may
be more convenient, we can, without trouble, add the developed formulas. The
general formulas will be
p sin {B — A) sin [H— P) = — h cos {H— A)-\-a cos [H— B)
p sin {B — A) cos {H— P) = h sin {H— ^) — « sin {H— B) .
Thus for 11=^ A, they change into
102 RELATIONS BETWEEN SEVERAL [BoOK I.
• I A Ti\ a COS (5 — A) — b
pBm{A-P)= ,.,\^_^)
pcos{A — P) = a.
For ff= B, they acquire a similar form ; but for ^= \{A-\-B) they become
;,sin(M+Ji?-P) = ,-^4^
^cos(M+ii?-P) = ,4y_^),
SO that the auxiliary angle t being introduced, of which the tangent =p it
becomes
tan (M + J ^ — P) = tan (C — 45°) cotan \{B — A).
Finally, if we desire to determine jt? immediately from a and h without previ-
ous computation of the angle P, we have the formula
p m^[B — A) = sj {aa'-\-ll — 1 al (^Q%{B — A)),
as well in the present problem as in 11.
79.
For the complete determination of the conic section in its plane, three things
are required, the place of the perihehon, the eccentricity, and the semi-parameter.
If these are to be deduced from given quantities depending upon them, there
must be data enough to be able to form three equations independent of each
other. Any radius vector whatever given in magnitude and position furnishes
one equation : wherefore, three radii vectores given in magnitude and position are
requisite for the determination of an orbit ; but if two only are had, either one
of the elements themselves must be given, or at all events some other quantity,
with which to form the third equation. Thence arises a variety of problems
which we will now investigate in succession.
Let r, /, be two radii vectores which make, with a right line drawn at pleasure
from the sun in the plane of the orbit, the angles N, N', in the direction of the
motion ; further, let 77 be the angle which the radius vector at perihehon makes
with the same straight line, so that the true anomaHes N — IT, N' — 77 may
answer to the radii vectores r, / ; lastly, let e be the eccentricity, and p the semi-
parameter. Then we have the equations
Sect. 3.] places in oebit. 103
^==l_^ecos(i\^— 77)
^=l-f gcos(i\^' — 77),
from wMch, if one of the quantities p, e, 77, is also given, it will be possible to
determine the two remaining ones.
Let us first suppose the semi-parameter p to be given, and it is evident that
the determination of the quantities e and 77 from the equations
ecos(i\^— 77) = ^— 1
gcos(i\r'_77) = ^— 1,
can be performed by the rule of lemma m. in the preceding article. We have
accordingly
tan {N- 77) ^ cotan {N'-N) - , (^4^-g,_^ •
tan {kNJr^N'-n) = ^^- '^ '"^'^'^ ^ fj^.
80.
If the angle 77 is given, p and e will be determined by means of the equations
rr' (cos {N— TI) — cos {N' — IT))
p
r cos (iV— n) — r' cos (N' — 77)
r cos {N — 77) — r' cos {K' — .77) *
It is possible to reduce the common denominator in these formulas to the form
a cos {A — 77), so that a and A may be independent of IT. Thus letting H de-
note an arbitrary angle, we have
r cos(iV— 77)— /cos (i\^'— 77) = (rcos (i\^— .S")— /cos(i\^'— 5^)) cos(^— 77)
— (r sin [N—H) —/sin {N'—H)) sin {R—U)
and so
= a cos (A — 77),
if a and A are determined by the equations
rcos{JV—R) — rcos{N' — B:)=acos{A — ff)
104 RELATIONS BETWEEN SEVERAL [BoOK I.
In this way we have
_2r'/sini(N'~IP)s\n(^N'Jri^'—n)
-^ a cos (J. — IT)
a cos {A — J7) ■
These formulas are especially convenient when p and e are to be computed for
several values of 11 ; r, r, N, N' continuing the same. Since for the calculation
of the auxiliary quantities a, A, the angle H may be taken at pleasure, it will be
of advantage to put H=^ h (-^-|- J^'), by which means the formulas are changed
into these, —
(/ _ r) cos ^{N' — N) = — a cos (.1 — h N— h N')
{r' -f r) sin ^{N' — N) = — a sin {A — h N— h iSf'),
And so the angle A being determined by the equation
tan {A — h N— i N') = ^^ tan ^ {W' — iV) ,
we have immediately
_ cos (A — I IT— jN')
^ cos i (xV — N) cos (A—n) '
im c
method already frequently explained
The computation of the logarithm of the quantity -p^- may be abridged by a
81.
If the eccentricity e is given, the angle 77 will be found by means of the
equation
C0S(7l 11)— ecos^iN'~W) '
afterwards the auxiliary angle A is determined by the equation
tan (A — i N— i N') = ^I^ tan i {N' — N)!
The ambiguity remaining in the determination of the angle A — 77 by its cosine
is founded in the nature of the case, so that the problem can be satisfied by two
different solutions ; which of these is to be adopted, and which rejected, must be
decided in some other way ; and for this purpose the approximate value at least
Sect. 3.] places in orbit. 105
of n must be already known. After IT is found, ^; will be computed by the
formulas
JO == /■ (1 -f e cos (iV^— 77)) = / (1 + e cos (iV^' — 77)),
or by this,
2 >•/ e sin ^ {N' — N) sin (^ N'^^ N— 11)
82.
Finally, let us suppose that there are given three radii vectores r, r, r", which
make, with the right line drawn from the sun in the plane of the orbit at pleasure,
the angles iV, iV', iV"". We shall have, accordingly, the remaining symbols being
retained, the equations
(L) l = lJ^eco8 {H— IT)
f=:l+.cos(iV^"-77),
from which p, IT, e, can be derived in several different ways. If we wish to
compute the quantity jo before the rest, the three equations (I.) maybe multiphed
respectively by sin {N" — N'\ — sin(iV" — N\ sin [N' — N\ and the products
being added, we have by lemma I., article 78,
sin {N" — N') — sin {N" — N)^ sin {W — N)
p:
\mi{N"—N') — ymi{N"—N)^~mi{N' — N)
This expression deserves to be considered more closely. The numerator evidently
becomes
2 sin h {N" — N') cos h {N" — N') — 2mi h {N" — N') cos {^ N" -\- h N' — N)
= 4 sin ^ {N" — N') sm i {N" — N) sin h {N' — N).
Putting, moreover,
r'r"%m{N" — N')=n, rr" Bm{N" — N)=^n', rr'Bm{N' — N) = n",
it is evident that h n, h n' i n", are areas of triangles between the second and thu-d
radius vector, between the first and third, and between the first and second.
14
106 RELATIONS BETWEEN SEVERAL [BoOK I.
Hence it will readily be perceived, that in the new formula,
__ 4sin 1 {N'" — N') sin ^ (¥" — J^) sin ^ (J^' — IP) . r / /'
" n — n' -\- n"
the denominator is double the area of the triangle contained between the ex-
tremities of the three radii vectores, that is, between the three places of the
heavenly body in space. When these places are httle distant from each other,
this area will always be a very small quantity, and, indeed, of the third order,
if N' — N, N" — N' are regarded as small quantities of the first order. Hence
it is readily inferred, that if one or more of the quantities r, r, r", N, N', N", are
affected by errors never so slight, a very great error may thence arise in the de-
termination of j» ; on which account, this manner of obtaining the dimensions of
the orbit can never admit of great accuracy, except the three heliocentric places
are distant from each other by considerable intervals.
As soon as the semi-parameter j» is found, e and 77 will be determined by the
combination of any two whatever of the equations I. by the method of article 79.
83.
If we prefer to commence the solution of this problem by the computation
of the angle 77, we make use of the following method. From the second of
equations I. we subtract the third, from the first the third, from the first the sec-
ond, in which manner we obtain the three following new equations : —
1 1
sm(iN-\-iN" — n)
Any two of these equations, according to lemma H., article 78, will give 77 and -,
whence by either of the equations (I.) will be obtained likewise e and p. If we
select the third solution given in article 78, II., the combination of the first equa-
1
1
2
^m\{N"
—
N') -
1
r
1
2
&m^{N"
' —
-V)—.
1
r
1
V
Sect. 3.] places in orbit. 107
tion with the third gives rise to the following mode of proceeding. The auxil-
iary angle C may be determined by the equation
tanc,— ^. gin i (jv^/ _ j^)
and we shall have
tan {i ]\r-\-iJSr'-]- i N" — n) = tan (45° + C) tan I {]V" — JV).
Two other solutions wholly analogous to this will result from changing the second
place with the first or third. Since the formulas for - become more complicated
by the use of this method, it will be better to deduce e and;;, by the method of
article 80, from two of the equations (I.). The uncertainty in the determination
of IT by the tangent of the angle i JY -\- ^ JV' -{- i JY'' — 77 must be so decided
that e may become a positive quantity: for it is manifest that if values 180° dif-
ferent were taken for 7/, opposite values would result for e. The sign of ]j, how-
ever, is free from this uncertainty, and the value of p cannot become negative,
unless the three given points lie in the part of the hyperbola away from the sun,
a case contrary to the laws of nature which we do not consider in this place.
That which, after the more difficult substitutions, would arise from the appli-
cation of the first method in article 78, IT., can be more conveniently obtained in
the present case in the following manner. Let the first of equations II. be multi-
plied by cos i {N" — N'), the third by cos h {N' — N\ and let the product of
the latter be subtracted from the former. Then, lemma I. of article 78 being
properly applied,* will follow the equation
* {v-v) ««t^^ " {N"-N') - h {\-\) cotan \ {JST'-JST)
= - sin I (i\r"_i\r) cos (I i\^-f i N" — n).
By combining which with the second of equations 11. 77 and - will be found ; thus,
77 by the formula
'Putting, that is, in the second formula, A=z\ {N" — N'), B=\N-{-\N"— 77, 0=i {N—N').
108 RELATIONS BETWEEN SEVERAL [BoOK I.
(l _^) cotan ^ {N" — N) — (^- — l) cotan h {N' — N)
Hence, also, two other wholly analogous formulas are obtained by interchanging
the second place with the first or third.
84.
Since it is possible to determine the whole orbit by two radii vectores given
in magnitude and position together with one element of the orbit, the time also
in which the heavenly body moves from one radius vector to another, may be
determined, if we either neglect the mass of the body, or regard it as known :
we shall adhere to the foraier case, to which the latter is easily reduced. Hence,
inversely, it is apparent that two radii vectores given in magnitude and position,
together with the time in which the heavenly body describes the intermediate
space, determine the whole orbit. But this problem, to be considered among the
most important in the theory of the motions of the heavenly bodies, is not so
easily solved, since the expression of the time in terms of the elements is tran-
scendental, and, moreover, very complicated. It is so much the more worthy of
being carefully investigated ; we hope, therefore, it will not be disagreeable to
the reader, that, besides the solution to be given hereafter, which seems to leave
nothing further to be desired, we have thought proper to preserve also the one
of which we have made frequent use before the former suggested itself to me.
It is always profitable to approach the more difiicult problems in several ways,
and not to despise the good although preferring the better. We begin with ex-
plaining this older method.
We will retain the symbols r, /, N, N', p, e, U with the same meaning, with
which they have been taken above ; we will denote the difference N' — N by J,
and the time in which the heavenly body moves from the former place to the
Sect. 3.] places in orbit. 109
latter by t Now it is evident that if the approximate value of any one of the
quantities p, e, U, is known, the two remaining ones can be determined from them,
and afterwards, by the methods explained in the first section, the time corre-
sponding to the motion from the first place to the second. If this proves to be
equal to the given time t, the assumed value o^ p, e, or 77, is the true one, and the
orbit is found ; bi^t if not, the calculation repeated with another value differing a
little from the first, will show how great a change in the value of the time corre-
sponds to a small change in the values of jo, e-, U; whence the correct value will
be discovered by simple interpolation. And if the calculation is repeated anew
with this, the resulting time will either agree exactly with that given, or at least
differ very little from it, so that, by applying new corrections, as perfect an agree-
ment can be attained as our logarithmic and trigonometrical tables allow.
The problem, therefore, is reduced to this, — for the case in which the orbit is
still wholly unknown, to determine an approximate value of any one of the quan-
tities p, e, 77. We will now give a method by which the value of p is obtained
with such accuracy that for small values of J it wul require no further correc-
tion ; and thus the whole orbit will be determined by the first computation with
all the accuracy the common tables allow. This method, however, can hardly
ever be used, except for moderate values of J, because the determination of
an orbit wholly unknown, on account of the very intricate complexity of the
problem, can only be undertaken with observations not very distant from each
other, or rather with such as do not involve very considerable heliocentric
motion.
Denoting the indefinite or variable radius vector corresponding to the true
anomaly v — 77 by (), the area of the sector described by the heavenly body in
the time t will be hf^ ^ d v, this integral being extended from v = iV to v = N',
and thus, (k being taken in the meaning of article 6), /c^ s} p =A^ ^ dv. . Now it
is evident from the fomulas developed by Cotes, that ii (fx expresses any
function whatever of x, the continually approximating value of the integral
/(fx .dx taken from x =: u to x = u -\- J is given by the formulas
110 RELATIONS BETWEEN SEVERAL [BotK i.
l-J((pu-\-(p{u-^ J))
i z/ (9) ?( + 3 9> (m -f 1 z/) 4- 3 9 (m + f //) -|- 9 (m + J)), etc.
It will be sufficient for our purpose to stop at the two first formulas.
By the first formula we have in our problem,
if we put
J ^ ^ ^ ' '' cos 2 w
— = tan (45° -[- ft>).
Wherefore, the first approximate value of \^p, which we will put = Sa, will be
/ Arr' o
' -^ k < cos 2 0)
By the second formula we have more exactly
fq ^ d v = i J (r r -J- r'r -|- 4 ^ i?) ,
denoting by R the radius vector corresponding to the middle anomaly
Now expressing p by means of r, R,r,N,N-\-k/l,N-\-J according to the for
mula given in article 82, we find
4 sin^ ^ ^ sin ^ z/
and hence
cos-^^ ^(}_^ 1 \ 2 sin^ \ A cos oj 2 sin^J^
R \T ~T~ "7/ 'p \j {r r cos 2(a) P '
"By putting, therefore,
2 sin^ ^ A ^ (r r' cos 2 w) «.
cos w '
we have
rf COS 1 J\J {r r' cos 2 ca)
cosw(l )
whence is obtained the second approximate value of y//>,
Sect. 3.] places m orbit. Ill
, I 2 a cos^ i A cos^ 2 a) ,
if we put
fi /cos 4 ^ COS 2 w\2
2a( — ^ 1 =e.
\ cos w /
"Writing, therefore, tt for \/jt?, tt will be determined by the equation
which properly developed would ascend to the fifth degree. We may put
71 = q -{- fi, so that q is the approximate value of jt, and fi a very small quantity,
the square and higher powers of which may be neglected : from which substitu-
tion proceeds
or
■■_gg'— (gg— «g)(gg— g)"
and so
sq'-\-(qq — 8)(aqq-\-4.8q—5ad)q
{qq—d){q^-^3 8q — 4:ad)
Now we have in our problem the approximate value of tt, namely, 3 a, which
being substituted in the preceding formula for q, the corrected value becomes
243a*s-\-3a{9aa~8) {9aa-}-78)
" (9aa— 8) (27 aa-{-5 8)
Putting, therefore,
-^-—3 ' —r
27aa~^' (1 — 3/3)a~~'»
the formula assumes this form,
ail±r±21f)
^- 1 + 5^ '
and all the operations necessary to the solution of the problem are comprehended
in these five formulas : —
I. - = tan (45° -\- (o)
112 RELATIONS S BETWEEN SEVERAL [BoOK 1.
m.
3 A < cos 2 oj
2 sin'^ \A^{rr' cos 2 w)
21 a a cos o)
j^ 2 cos^^ Acqs'' 2 at
^^- (l_3/3)cos2w ~'^
V- i_|_5j3 — Vi^-
If we are willing to relinquish something of the precision of these fonnulas, it
will be possible to develop still more simple expressions. Thus, by making cos w
and cos 2 w = 1, and developing the value of \j p m. n. series proceeding according
to the powers of//, the fom-th and higher powers being neglected, we have,
in which /i is to be expressed in parts of the' radius. Wherefore, by making
we have
VI. p=j,'{i-ijj + 4^).
In like manner, by developing \^ p in a series proceeding according to the powers
of sin J, putting
r / sin A i //
we have
or
YKL p =/' + ^ sin^ Jsjrr.
The formulas VII. and VlLl. agree with those which the illustrious Euler has
given in the Theoria moius phneianmi et comdanm, but formula VI., with that which
has been introduced in the Recherches et cakiils sur la vraie orbite ellipiique de la
cmiete de 1769, p. 80.
Sect. 3.]
PLACES m ORBIT.
113
87.
The following examples will illustrate the use of the preceding precepts, while
from them the degree of precision can be estimated.
I. Let log r = 0.3307640, log / = 0.3222239, z/ = 7° 34' 53".73 = 27293".73,
t= 21.93391 days. Then is found m = — 33'47".90, whence the further compu-
tation is as foUows : —
log J . .
logrr . .
C.log3^ . .
CAogt. . .
C. log cos 2 0)
4.4360629
0.6529879
5.9728722
8.6588840
0.0000840
1 log y. / QQQ 2 0)
2 log sin i J
log-ij . .
C. log a a
C. loe; cos w .
log« . . . . 9.7208910
0.3264519
7.0389972
8.8696662
0.5582180
0.0000210
log|5 6.7933543
8= 0.0006213757
log2 . . . .
2 log cos i J
2 log cos 2 0)
Clog (1-3/5)
2 C. log cos w
0.3010300
9.9980976
9.9998320
0.0008103
0.0000420
1 + / + 21/3
3.0074471
log
log« . . . .
Clog (1 + 5^)
0.4781980
9.7208910
9.9986528
logr . . . . 0.2998119
y = 1.9943982
21 /5 == 0.0130489
logy/i?
log^
0.1977418
0.3954836
This value of log p differs from the true value by scarcely a single unit in the
seventh place: formula VI., in this example, gives log j» = 0.3954822; formula
Vn. gives 0.3954780 ; finally, formula Vm., 0.3954754.
n. Let log r= 0.4282792, log /= 0.4062033, // = 62° 55' 16".64, if =259.88477
days. Hence is derived co = — 1° 27'20".14, log a = 9.7482348, /? = 0.04535216,
y = 1.681127, log s/ p = 0.2198027, logj^ == 0.4396054, which is less than the true
value by 183 units in the seventh place. For, the true value in this example is
0.4396237 ; it is found to be, by formula VI, 0.4368730 ; from formula VH. it
15
114 EELATIONS BETWEEN SEVERAL [BoOK I.
results 0.4159824 ; lastly, it is deduced from formula YIH., 0.4051103 : the two
last values differ so much from the truth that they camiot even be used as ap-
proximations.
The exposition of the second method will afford an opportunity for treating
fully a great many new and elegant relations ; which, as they assume different
forms in the different kinds of conic sections, it will be proper to treat separately ;
we will begin with the ELLIPSE.
Let the eccentric anomalies U, E', and the radii vectores r, r, correspond to
two places of the true anomaly v, v', (of which v is first in time) ; let also p
be the semi-parameter, e = sin 9) the eccentricity, a the semi-axis major, t the
time in which the motion from the first place to the second is completed ; finally
let us put
v' — v = 2f, v' ^v = 1F, E' — E=z2g, E'-\-E=2G, acoscp = ^ = b.
Then, the following equations are easily deduced from the combination of for-
mulas v., VI., article 8 : —
[1] ^ sin ^ = sin/, y/ r /,
[2] bsma = smF.\/rr,
jt? cosy = (cos I ^' COS I ^;^ (1 -|- e) -[- sin ^ y sin ^ v'. (1 — e)) y/ r r, or
[3] p cosy = (cos/-j- e cos F) y/ r r, and in the same way,
[4] JO cos 6^ = (cos J' 4" ^ cos/) y/r/.
From the combination of the equations 3 and 4 arise,
[5] cos/. y/r/ = (cosy — e cos G) a,
[6] cosi^.y/r/= (cos 6^ — e cosy) a.
From formula III., article 8, we obtain
[7] / — r = 2 ae siny sin (r,
r -\-r =^2 a — 2ae cosy cos G =i2a sin^y -\- 2 cos/cosyy/r/j
whence,
[8] a
r -\-r' — 2 cos/cos^'y/r/
Sect. 3.] places m oebit. 115
Let us put
[9] \/f+/7^
2 cos/ ' '
and then will
rlQ■^ ^ _ ^ (^ + sin^iff) cos/y/r/,
also
' — sin^ '
in which the upper or lower sign must be taken, as sin y is positive or negative.
Formula XII., article 8, furnishes us the equation
k t
— =^E' — esin^' — ^-|-esin^=2^ — 2esin^cos(?
= 2^ — sin2^-|-2 cos/sin^^^.
If now we substitute in this equation instead of a its value from 10, and put, for
,the sake of brevity,
[11] I 5=^j
2^cos/2'(r/)*
we have, after the proper reductions,
[12] +m = (^+sinn^)^+(^+sinn^)^f^-:^"^1.
in which the upper or lower sign is to be prefixed to m, as sin g is positive or
negative.
"When the heliocentric motion is between 180° and 360°, or, more generally,
when cos/ is negative, the quantity m determined by formula 11 becomes im-
aginary, and I negative ; in order to avoid which we will adopt in this case, instead
of the equations 9, 11, the following: —
1 [9=^] VV+V/_i_2^
■- -■ 2 cos/ '
[11*] -1 '-^^ ,=M,
2^(— cos/f (r/f
whence for 10, 12, we shall obtain these, —
116 RELATIONS BETWEEN SEVERAL [BoOK I.
nn:in — 2 (Z— sin^lgr) cos/y/r/
[-12.] +Jf^_(i:_sinn^)'+(Z-sin^^^)^-(^lr^^)^
in which the doubtful sign is to be determuied in the same manner as before.
89.
"We have now two things to accompHsh ; first, to derive the unknown quan-
tity (/ as conveniently as possible from the transcendental equation 12, since it
does not admit of a direct solution ; second, to deduce the elements themselves
from the angle (/ thus found. Before we proceed to these, we will obtain
a certain transformation, by the help of which the computation of the auxiliary
quantity I ov L is more expeditiously performed, and also several formulas after-
wards to be developed are reduced to a more elegant form.
By introducing the auxihary angle w, to be determined by means of the
formula
''4 = tim(45° + <»),
^
we have
J^-\- sjy = 2 -f (tan (45° + t«) — cotan (46° -f w))^ = 2 + 4 tan^ 2 w;
whence are obtained
, sia^l/ I tan2 2a) ^ sin^i/ tan'^ 2 to
cos/ "T" cos/ ' cos/ cos/
90.
"We will consider, in the first place, the case in which a value of </ not very
great, is obtained from the solution of the equation 12, so that
2ff — sm2g
sin^ g
may be developed in a series arranged according to the powers of sin J (/. The
numerator of this expression, which we shall denote by X, becomes
-\2- sin^ h (/ — -\-^- sin^ ^ y — | sin' ^ (/ — etc. ;
Sect. 3.] ' places m oebit. 117
and the denominator,
8 sin^ i^ — 12 sin^ a ^ + 3 sin^ 2 (/ -\- etc.
Whence X obtains the form
I +1 sin^ i^ + II sinn^+ etc.
But in order to obtain the law of progression of the coefficients, let us differen-
tiate the equation
X sin^^ = 2 y — sin 2 ^,
whence results
3Xcos^sin^^-f-^i^^.^x^= 2 — 2 cos 2^ = 4:sin^^;
putting, moreover,
sin^ h g=^x,
"We have
^g = Hmg,
whence is deduced
dX 8 — eXcos^ 4 — 3X(1— 2a;)
da; sin^^ 2x{l — x) '
and next,
{%x — 2xx)~ = i — {^ — %x)X.
If, therefore, we put
X= ^{l^ax -^ ^ XX ^ r ci^ -\-d x''^ Qic.)
we obtain the equation
^(ax-{-{2§ — a)xx^{^r — 2^)x^-\-{id — ^y)x'^eiQ?}
=.{^-^4.a)x-\-{9>a — 4.^)xx-\-{^§ — 4.r)x^^{%r — 4.d)x^^Qic.
which should be identical. Hence we get
in which the law of progression is obvious. We have, therefore,
j^_4 I ^-e^ I ^-e-S I 4.6.8.10 3 ■ 4.6.8.10.12 . ■
3 -t-3, 5^-1-3. 5. 7^^-1-3. 5. 7, 9 ^ -h 3.5.7.9.11 ^^ e^c.
This series may be transformed into the following continuous fraction : —
118 RELATIONS BETWEEN SEVERAL ' [BoOK 1.
1 I 2
1 + 577^
T 5.8
1-779^
i-9TTr^
1 7.10
-^~1LT3^
-^""15.17^
1 — etc.
The law according to which the coefficients
6 2_ 5^ h£
5' ~577' 779' 9^1' ^ ^'
proceed is obvious; in truth, the n"" term of this series is, when n is even.
2»i + 1.2rt + 3'
when n is odd,
n-\-2.n-\-o
2n-\-1.2n/-}-3'
the further development of this subject would be too foreign from our pm'pose.
If now we put
1-
'7.9
X
1-
1.
■9T
.4
IT
X
we have
1
—
etc.
Sect. 3.] PLACES IN OKBIT. 119
and
or
sm^g — ^(2 g — sm2 g) (l—ism^g)
The numerator of this expression is a quantity of the seventh order, the denomi-
nator of the third order, and ^, therefore, of the fourth order, if ^ is regarded as
a quantity of the first order, and x as of the second order. Hence it is inferred
that this formula is not suited to the exact numerical computation of ^ when ^
does not denote a very considerable angle: then the following formulas are
conveniently used for this purpose, which differ from each other in the changed
order of the numerators in the fractional coefiicients, and the first of which is
derived without difficulty from the assumed value of x — 5.*
[13] ^
1+
A
X-
-u
X
X
m
1-
-■h
X
1—
X
etc.,
^ =
■sS
1 — il^— 6¥^
1 4.O..
iVs^
JO X
1 — etc.
In the third table annexed to this work are found, for all values of x from
0 to 0.3, and for every thousandth, corresponding values of \ computed to
seven places of decimals. This table shows at first sight the smallness of 1 for
* The derivation of the latter supposes some less obvious transformations, to be explained on another
occasion.
120 RELATIONS BETWEEN SEVERAL [BoOK I.
moderate values of g ; thus, for example, for E' — ^=10°, or y = 5°, when
X = 0.00195, is ^ = 0.0000002. It would be superfluous to continue the table fur-
ther, since to the last term a;=0.3 corresponds g = 66° 25', or U' — U=^ 132° 50'.
The third column of the table, which contains values of ^ corresponding to nega-
tive values of x, will be explained further on in its proper place.
91.
Equation 12, in which, in the case we are treating, the upper sign must evi-
dently be adopted, obtains by the introduction of the quantity ^ the form
Putting, therefore,
and
[14]
t+z+§ — -' 0
the proper reductions being made, we have
[15] A = fc^^
If, accordingly, h may properly be regarded as a known quantity, f/ can be de-
termined from it by means of a cubic equation, and then we shaU have
[16] x = "^ — l.
Now, although h involves the quantity |, still unknown, it will be allowable to
neglect it in the first approximation, and for h to take
m m
since ^ is undoubtedly a very small quantity in the case we are discussing.
Hence ,y and z will be deduced by means of equations 15, 16 ; | will be got
from X by table m., and with its aid the corrected value of h will be obtained by
formula 14, with which the same calculation repeated will give corrected values
of y and x : for the most part these will differ so little from the preceding, that |
Sect. 3.] places in orbit. 121
taken again from table III., will not differ from the first value ; otlierAvise it ^vould
be necessary to repeat the calculation anew until it underwent no further change.
When the quantity x shall be found, g will be got by the formula sin^ 2 y = x.
These precepts refer to the first case, in which cos/ is positive \ in the other
case, where it is negative, we put
v/(X-:.) = 5
and
["*] x^^.=^.
whence equation 12* properly reduced passes into this,
[15.] B=(^%YJ-
Y and H can be determined, accordingly, by this cubic equation, whence again x
wiU be derived from the equation
[16*] x = L — -^^.
In the first approximation
MM
will be taken for H; \ will be taken from table m. with the value of x derived
from H by means of the equations 15*, 16*; hence, by formula 14*, will be had
the corrected value of H, with which the calculation will be repeated in the same
manner. Finally, the angle g will be determined from x in the same way as in
the first case.
92.
Although the equations 15, 15*, can have three real roots in certain cases, it
will, notwithstanding, never be doubtful which should be selected in our problem.
Since h is evidently a positive quantity, it is readily inferred from the theory
of equations, that equation 15 has one positive root with two imaginary or two
negative. Now since
16
122 ■ RELATIONS BETWEEN SEVERAL [BoOK 1.
must necessarily be a positive quantity, it is evident that no uncertainty remains
here. So far as relates to equation 15*, we observe, in the first place, that L is
necessarily greater than 1 ; which is easily proved, if the equation given in article
89 is put under the form
r- 1 I cos^ 2/1 ^"^^ 2 (a .
"^ — cos/ ' — cos/'
Moreover, by substituting, in equation 12*, YsJ [L — x) in the place of 31, we
have
Y^\ = {L-x)X,
and so
r+i>(i-:.)X>| + A^ + AL^^_l_A^^4_etc.>|,
and therefore F^ |^. Putting, therefore, Y=l -\- Y', Y' wiU necessarily be a
positive quantity ; hence also equation 15* passes into this,
which, it is easily proved from the theory of equations, cannot have several posi-
tive roots. Hence it is concluded that equation 15* would have only one root
greater than i,-|- which, the remaining ones being neglected, it will be necessary
to adopt in our problem.
93.
In order to render the solution of equation 15 the most convenient possible
in cases the most frequent in practice, we append to this work a special table
(Table II.), which gives for values of h from 0 to 0.6 the corresponding loga-
rithms computed with great care to seven places of decimals. The argument
h, from 0 to 0.04, proceeds by single ten thousandths, by which means the
second differences vanish, so that simple interpolation suffices in this part
of the table. But since the table, if it were equally extended throughout,
would be very voluminous, from h = 0.04 to the end it was necessary to proceed
by single thousandths only ; on which account, it will be necessary in this latter
part to have regard to second differences, if we wish to avoid errors of some units
t If in fact we suppose that our problem admits of solution.
Sect. 3.] places in orbit. , 123
in the seventh figure. The smaller values, however, of Ji are much the more fre-
quent in practice.
The solution of equation 15, when h exceeds the limit of the table, as also
the solution of 15*, can be performed without difficulty by the indirect method,
or by other methods sufficiently known. But it will not be foreign to the pur-
pose to remark, that a small value of g cannot coexist with a negative value of
cos/, except in an orbit considerably eccentric, as will readily appear from equa-
tion 20 given below in article 95."}*
94.
The treatment of equations 12, 12* explained in articles 91, 92, 93, rests upon
the supposition that the angle g is not very large, certainly within the limit 66° 25',
beyond which we do not extend table III. When this supposition is not correct,
these equations do not require so many artifices ; they can be most securely
and conveniently solved by trial tvithout a change of form. Securely, since the value
of the expression
2 g — sin 2 ^r
sin* 5- '
in which it is evident that 2g is to be expressed in parts of the radius, can, for
greater values of ^,be computed with perfect accuracy by means of the trigonomet-
rical tables, which certainly cannot be done as long as ^ is a small angle : con-
veniently, because heHocentric places distant from each other by so great an interval
will scarcely ever be used for the determination of an orbit wholly unknown, while
by means of equation 1 or 3 of article 88, an approximate value of g follows
with almost no labor, from any knowledge whatever of the orbit: lastly, from an
approximate value of g, a corrected value will always be derived with few trials,
satisfying with sufficient precision equation 12 or 12*. For the rest, when two
given heliocentric places embrace more than one entire revolution, it is necessary
to remember that just as many revolutions will have been completed by the eccen-
tric anomaly, so that the angles^' — E, v' — v, either both lie between 0 and 360°,
t That equation shows, that if cos/ is negative, qo must, at least, be greater
than 90°— ^r.
124 RELATIONS BETWEEN SEVERAL [BoOK I
or both between similar multiples of the wbole circumference, and also / and g
together, either between 0 and 180°, or between similar multiples of the semicir-
cumference. If, finally, the orbit should be wholly unknown, and it should not
appear whether the heavenly body, in passing from the first radius vector to the
second, had described a part only of a revolution or, in addition, one entire revo-
lution, or several, our problem would sometimes admit several different solutions :
however, we do not dwell here on this case, which can rarely occur in practice.
95.
"We pass to the second matter, that is, the determination of the elements from
the angle g when found. The major semiaxis is had here immediately by the
formulas 10, 10*, instead of which the following can also be used : —
2 mm cos f\J rr' khtt
[17] a:
yysm^g 4 yy rr cos- fsin^g
[--, K:::-| — 2 MM COS fsj r v' kktt
The minor semiaxis h = ^ ap is got by means of equation 1, which being
combined with the preceding, there results
Now the elliptic sector contained between two radii vectores and the elliptic arc
is h kt \J p, also the triangle between the same radii vectores and the chord
^rr' sin 2/: wherefore, the ratio of the sector to the triangle is as^: 1 or Y: 1.
This remark is of the greatest importance, and elucidates in a beautiful manner
both the equations 12,12*: for it is apparent from this, that in equation 12 the
i. a. 1. 5
parts m, {l-{-xY, X(/-|-.r) , and in equation 12* the parts M, {L — xY,X{L — xy,
are respectively proportional to the area of the sector (between the radii vectores
and the elliptic arc), the area of the triangle (between the radii vectores and the
chord), the area of the segment (between the arc and the chord), because, the
first area is evidently equal to the sum or difference of the other two, accord-
ins; as v — V lies between 0 and 180°, or between 180° and 360°. In the case
Sect. 3.] places in orbit. 125
where v — v is greater than 360° we must conceive the area of the whole eUipse
added to the area of the sector and the area of the segment just as many times
as the motion comprises entire revolutions.
Moreover, since 5 =a cos 9, from the combination of equations 1, 10, 10*,
follow
n A-i sin q tan f
[19] cosq) = -,, , ■ 21 V
[19*] c«s9 = ^Si^.
whence, by substituting for I, X, their values from article 89, we have
rc\f\-y sin /"sin q
[20] coscp = - -^ — , ^, 2.0 •
i- -• ' 1 — COS/ COS 5^ -|- 2 tan'' 2 to
This formula is not adapted to the exact computation of the eccentricity
when the latter is not great : but from it is easily deduced the more suitable
formula
[21] tan^i^)
sinH(/— 5r)+tan2 2.
to which the following form can likewise be given (by multiplying the numerator
and denominator by cos^ 2 w)
[22] tan^ I 9
^ sin^ \ if—g) + cos^ \ (/- g) sin'' 2 a>
sinH {f-\-9) + cosH (/— 5") sin2 2 w*
The angle 9 can always be determined with all accuracy by either formula, using,
if thought proper, the auxiliary angles of which the tangents are
tan 2 CO tan 2 oi
sini(/— 5^)' sin^CZ+^r)
for the former, or
tani(/-^)' tani(/+^)
for the latter.
The following formula can be used for the determination of the angle G,
which readily results from the combination of equations 5, 7, and the following
one not numbered,
(/ — r)sin5r
[23] tan G = l^-^;sm^
■- -^ {r-\-r)cosg — 2, cos, jsjrr"
from which, by introducing w,is easily derived
126 RELATIONS BETWEEN SEVERAL [BoOK I.
rOA~\ + /^ sin ^ sin 2 ft)
L J cos''2ajsia-^(/ — 5^) sin^(/-|-^)-[-sin^2 tocos^''
The ambiguity here remaining is easily decided by means of equation 7, which
shows, that G must be taken between 0 and 180°, or between 180° and 360°,
as the nimierator in these two formulas is positive or negative.
By combining equation 3 with these, which flow at once from equation II.
article 8,
1 1 2e . . . r^
r r p ''
1,1 2 , 2e . J,
- + -J- = COS / cos I',
r ' / P P
the following will be derived without trouble,
F251 tani^= (/ — y)sin/
L -I 2 cos g^rr' — (r'-\-r)cosf'
from which, the angle la being introduced, results
r9fin + TP sin /sin 2 to
•- -^ cos^ 2 tu sin i- (/ — g) sia^ {f-\-g) — sin^2(«cos/*
The uncertainty here is removed in the same manner as before. — As soon as
the angles F and Cr shall have been found, we shall have v = F — /, v' = F-\-f,
whence the position of the perihelion will be known ; also E= G — g, E'^G -\-g.
Finally the mean motion in the time t will be
— ^=2g — 2 ecosG^sin^,
a*
the agreement of which expressions will serve to confirm the calculation ; also,
the epoch of the mean anomaly, corresponding to the middle time between the
two given times, will be G — esinG cos ^, which can be transferred at pleasure
to any other time. It is somewhat more convenient to compute the mean
anomalies for the two given times by the formulas E — emiE, E' — e sin E', and
to make use of their difference for a proof of the calculation, by comparing it with
S
Sect. 3.] places in oebit. , 127
96.
The equations in the preceding article possess so much neatness, that there
may seem nothing more to be desired. Nevertheless, we can obtain certain
other formulas, by which the elements of the orbit are determined much more
elegantly and conveniently; but the development of these formulas is a little
more abstruse.
We resume the following equations from article 8, which, for convenience, we
distinguish by new numbers : —
I. smivJ^ = sm^U^{l-\-e)
n. cos|yi/^=cos^^\/(l — e)
m. sin^/y/^=sin^^V(l + «)
We multiply I. by sin h {F-\-g), II. by cos h {F-[-g), whence, the products bemg
added, we obtain
or, because
v/ (1 -[- e) = cos i 9 -j- sin I 9, \/ (1 — e) = cos ^ 9 — sin I 9,
cos h {f-{-g) \Jl = cos i 9 cos (i #— ia-\-g) — sm i if go& h {F -\- G) .
In exactly the same way, by multiplying HI. by sin h {F — g), IV. by cos I {F — g),
the products being added, appears
cos i {f-\-g) ^^ = cos icp cos{^F— ^G—g) — sin. | 9 cos h {F-\-G).
The subtraction of the preceding from this equation gives
cos i {f-j-g) (^-^_y/-l) = 2 COS i 9 sm^ sin ^ {F— G),
or, by introducing the auxiliary angle w,
[27] cos h if-^-g) tan 2 to = sin i {F—G) cos h 9 sin^ ^l^.
128 RELATIONS BETWEEN SEVERAL [BoOK 1.
By transformations precisely similar, the development of which we leave to the
skilful reader, are found
[28] '^^ = <'Osi{F-G)cosi<i,.mg^''^,
[29] cos h (/ — y) tan 2 to = sin ^ (i^-|- 6^) sin ^ 9 sin^ t/ —,,
[30] S^^i^==cos}(-P+ff)smJ9sm^^?^.
When the first members of these four equations are known, ^ (F — G) and
1 • V«« 7-»
cosi9)smyy/^ = P
will be determined from 27 and 29 ; and also, from 29 and 30, in the same manner,
^(^+6^) and
the doubt in the determination of the angles ^ {F — G), i {F-\-G), is to be so
decided that P and Q may have the same sign as siny. Then ^ cp and
•will be derived from P and Q. From P can be deduced
EEJr/
« = — r^ ,
and also
unless we prefer to use the former quantity, which must be
+ V (2 {l-\- sin^ hg) cos/) =+ y/ (— 2 (P — sin^ i^) cos/),
for a proof of the computation chiefly, in which case a and p are most conven-
iently determined by the formulas
7 sinfJr'/ b ,
0 = — "^ — , a= , » = ocoscp.
sin ^ ' cos (J) ' -^ '
Several of the equations of articles 88 and 95 can be employed for proving the
calculation, to which we further add the following; : —
2 tan 2 to / rr'
cos 2cu
v/^^^^^"^^^"^^
Sect. 3.]
PLACES IN ORBIT.
129
2 tan 2 to I pp
2 tan 2 0)
tan (f sin G sin/^ tan cp sin ^sin^.
cos 2 CO
Lastly, the mean motion and the epoch of the mean anomaly will be found in the
same manner as in the preceding article.
97.
We will resume the two examples of article 87 for the illustration of the
method explained in the 88th, and subsequent articles : it is hardly necessary to
say that the meaning of the auxiliary angle w thus far adhered to is not to be
confounded with that with which the same symbol was taken in articles 86, 87.
I In the first example we have /= 3° 47' 26".865, also
log ^ = 9.9914599, log tan (45° -\-oi) = 9.997864975, w = — 8' 27''.006.
Hence, by article 89,
log sin^ i /
log cos/ .
7.0389972
9.9990488
log tan^ 2 M
log cos/ .
5,3832428
9.9990488
7.0399484
= log 0.0010963480
and thus ^=0.0011205691, | -)-/= 0.8344539,
log Jet . . . . 9.5766974
5.3841940
= log 0.0000242211
Further we have
2 loff k t
C. log 8 cos^/
\ogmm
log(f + 0
9.1533948
9.0205181
9.0997636
7.2736765
9.9214023
7.3522742
The approximate value, therefore, of h is 0.00225047, to which in our table II.
corresponds \ogyy = 0.0021633. We have, accordingly,
log 7^7 = 7.2715132, or '"^^ 0.001868587,
yy yy '
17
130
RELATIONS BETWEEN SEVERAL
[Book I
whence, by formula 16, x = 0.000T480179 : wherefore, since | is, by table m.,
wholly insensible, the values found for h, y, x, do not need correction. Now, the
determination of the elements is as follows : —
log a; 6.8739120
logsin^<7 . . . 8.4369560, hg = rW 2".0286, ^ (/+<7) = 3° 27' 45^4611,
^ {f—g) = 19'41".4039. Wherefore, by the formulas 27, 28, 29, 30, is had
log tan 2 w . . . 7.6916214^2 Clog cos 2 w . . . 0.0000052
log cos ^(/+y)
log cos ^ (/—<?')
9.9992065
9.9999929
log sin ^(/-f-^)
log sin I (/—^)
log P sin ^(i^— 6^) 7.6908279^2
log P cos i {F— G) 8.7810240
log Q sin i (P4- G)
log^cosi(P-f 6^)
8.7810188
7.7579709
7.6916143 w
7.7579761
HF-G) =
— 4°38'4r.54
i,{FJrG) =
319 21 38 .05
F=
314 42 56 .51
v =
310 55 29 .64
v' =
318 30 23 .37
G=z
324 0 19.59
E =
320 52 15 .53
E'=
327 8 23.65
log P = log i? cos ^9 8.7824527
log ^ = log i^ sin h (f 7.8778355
Hence ^ cp = T & 0".935
9 = 14 12 1 .87
logP 8.7857960
For proving the calculation.
^ log 2 cos/. . . . 0.1500394
Hog(/+^) = log- 8.6357566
8.7857960
i log r r .
log sin/ .
Clog sin^
0.3264939
8.8202909
1.2621765
logb .
log cos cp
0.4089613
9.9865224
logp
log a
0.3954837
0.4224389
log sin (p .
log 206265
9.3897262
5.3144251
log e in seconds
log sin ^ . . .
log sin F' . .
4.7041513
9.8000767 «
9.7344714 w
log e sin ^
log e sin F'
4.5042280 w
4.4386227 «
Sect. 3.] places in orbit. 131
logk ... 3.5500066 esin^ == — 31932'a4 =— 8°52'12'a4
flog a . . . 0.6336584 gsin^'= — 27455 .08 =— 7 37 35 .08
2.9163482 Hence the mean anomaly for the
logt . . . 1.3411160 first place = 329°44'27".67
4 2574642 ^^^ ^^^ second = 334 45 58 .73
Difference = 5 1 31 .06
Therefore, the mean daily motion is 824".7989. The mean motion in the time
?; is 1809r.07 = 5° r3r.07.
II. In the other example we have
/=3r2r38''.32, w= — 2r50".565, J= 0.08635&59, logmm = 9.3530651,
-r-T-^, or the approximate value of h = 0.2451454 ;
to this, in table U., corresponds logj/y = 0.1722663, whence is deduced
"^ = 0.15163477, X = 0.06527818,
hence from table III. is taken ^ = 0.0002531. Which value being used, the cor-
rected values become
h = 0.2450779, logj/ j/ = 0.1722303, "^ = 0.15164737, x = 0.06529078,
^ = 0.0002532.
If the calculation should be repeated with this value of | , differing, by a single
unit only, in the seventh place, from the first; h, logy?/, and x would not suffer
sensible change, wherefore the value of x already found is the true one, and we
may proceed from it at once to the determination of the elements. We shall
not dwell upon this here, as it differs in nothing from the preceding example.
III. It will not be out of place, to elucidate by an example the other
case also in which cos/ is negative. Let v' — v = 224° 0' 0", or /= 112° 0' 0'',
log r = 0.1394892, log /= 0.3978794, ?;= 206.80919 days. Here we find
to z=: -f 4° 14'43'' 78, L = 1.8942298, log MM= 0.6724333, the first approximate
value of log^= 0.6467603, whence by the solution of equation 15* is obtained
¥= 1.591432, and afterwards x = 0.037037, to which, in table HI, corresponds
g = 0.0000801. Hence are derived the corrected values log ^= 0.6467931,
Y= 1.5915107, x= 0.0372195, £ = 0.0000809. The calculation being repeated
132 RELATIONS BETWEEN SEVERAL [BoOK L
with this value of I, we have x = 0.0372213, which value requires no further cor-
rection, since \ is not thereby changed. Afterwards is found h g =^ 11° 7'25'^40,
and hence ia the same manner as in example I.
^{F—G)=^ 3°33'63".59 log P = log i^ cos i 9 9.9700507
i(P+^)= 8 26 6.38 log ^ = logi?sini9 . 9.8580552
Fz= 1159 59.97 \(p= 37°41'34".27
y = _ 100 0 0 .03 9 = 75 23 8 .54
v'= +123 59 59.97 \ogR 0.0717096
^ ^= 40-iii:j.7y For proving the calculation.
E= —17 22 38.01 iog^y/_2cos/ . . 0.0717097
E'= +27 7 3.59 ^
The angle g) in such eccentric orbits is computed a little more exactly by
formula 19*, which gives in our example 9= 75° 23' 8''.57; likewise the eccen-
tricity e is determiaed with greater precision by the formula
g^l_2sin2(45° — ^9),
than by e ^ sin y ; according to the former, e = 0.96764630.
By formula 1, moreover, is found log b = 0.6576611, whence \ogp= 0.0595967,
log a = 1.2557255, and the logarithm of the perihelion distance
log j^ = \oga{l — e) = \ogb tan (45° — l^>) = 9.7656496.
It is usual to give the time of passage through the perihelion in place of the
epoch of the mean anomaly in orbits approaching so nearly the form of the
parabola ; the intervals between this time and the times corresponding to the
two given places can be determiaed from the known elements by the method
given in article 41, of which intervals the difference or sum (accordiag as the
perihelion lies without or between the two given places), since it must agree with
the time t, will serve to prove the computation. The numbers of this third ex-
ample were based upon the assumed elements in the example of articles 38, 43,
as indeed that very example had furnished our first place : the trifling differences
of the elements obtained here owe their origin to the limited accuracy of the
logarithmic and trigonometrical tables.
Sect. 3.] places in orbit. 133
The solution of our problem for the ellipse ia the preceding article, might be
rendered applicable also to the parabola and hyperbola, by considering the parab-
ola as an ellipse, in which a and l would be infinite quantities, ^ =^ 90°, finally
E, E', g, and 6^ = 0 ; and in a like manner, the hyperbola as an ellipse, in which a
would be negative, and h, E, E', g, G, (f, imaginary : we prefer, however, not to
employ these hypotheses, and to treat the problem for each of the conic sections
separately. In this way a remarkable analogy wiU readily show itself between
all three kinds.
Eetauiing in the PAEABOLA the sjrmbols p, v, v', F,f, r, r', i with the same sig-
nification with which they had been taken above, we have from the theory of the
parabolic motion : —
[1] ^Z.= cosn-^-/)
[2] sjl, = ^o^h{F^f)
^ = tan i {F +/) — tan ^ {F—f) -|- \ tan^ | ( J^ +/) _ | tan^ \ {F—f)
pi
= (tan i {F-]-f) — tan i {F—f)) (l + tan i {F-\-f) tan i {F—f) +
i (tan ^ {F-^f) — tan i {F—f)y)
2 sin/v/ r r' /2 cos fsjrr' .4: sin^ fr r'X
P \ P ' 3pp /'
whence
Further, by the multiplication of the equations 1, 2, is derived
[4] ^ = cosi^4-cos/
and by the addition of the squares,
[-5] ^4i^^ = l + cos^cos/.
134 RELATIONS BETWEEN SEVERAL [BoOK I.
Hence, cos F being eliminated,
rfil _ 2r/sinY
\y\ ^ — ^-|_/_2cos/Vr/*
If, accordingly, we adopt here also the equations 9, 9*, article 88, the first for
cos/ positive, the second for cos/ negative, we shall have,
which values being substituted in equation 3, preserving the symbols m, 31, with
the meaning established by the equations 11, 11*, article 88, there result
[8] m=l^-\-il^
[8*] jf=_x^4-|i:^.
These equations agree with 12, 12* article 88, if we there put ^= 0. Hence it is
concluded that, if two heliocentric places which are satisfied by the parabola, are
treated as if the orbit were elliptic, it must foUow directly from the application
of the rules of article 19, that a;= 0; and vice versa, it is readily seen that, if
by these rules we have a; = 0, the orbit must come .out a parabola instead of
an ellipse, since by equations 1, 16, 17, 19, 20 we should have b = co, a=co,
(p = 90. After this, the determination of the elements is easily efiected. Instead
of p, either equation 7 of the present article, or equation 18 of article 95 f might
be employed : but for F we have from equations 1, 2, of this article
tan ^ F=z ^ /T / cotan J / ^ sin 2 cy cotan h f,
if the auxiliary angle is taken with the same meaning as in article 89.
We further observe just here, that if in equation 3 we substitute instead of
p its value from 6, we obtain the well-known equation
kt = }{r-{-r -\-cosf.^ r/){r~\-r— 2 cos f.sjrr'y \/ 2.
t Whence it is at once evident that y and T express the same ratios in the parabola as in the
ellipse. See article 95.
Sect. 3.J places in orbit. 135
99.
We retain, in the HYPERBOLA also, the symbols jt?, v, v',f, F, r, r', t with the
same meaning, but instead of the major semiaxis a, which is here negative, we
shall write — a ; we shall put the eccentricity e = ^-^ in the same manner as
above, article 21, etc. The auxiliary quantity there represented by u, we shall
put for the first place =-, for the second =^ Oc. whence it is readily inferred
that c is always greater than 1, but that it differs less from one, other things
being equal, in proportion as the two given places are less distant from each
other. Of the equations developed in article 21, we transfer here the sixth and
seventh shghtly changed in form,
[1] eos|.= K\/T + \/^)\/^^
C2] sin|.:.K^^-^^)^^i±l)-«
[3] cos^.'=^(v/C'. + y/i^)y/(-^-'>"
[4] ,\nhv'=.h{sJOc-yJ^)s,l^^
From these result directly the following : —
[5] sin i^=^«(C- 1)^/1^
[6] ,inf^ha{c-])sj'-^
[8] cos/ = (.(C + i)-(.4-i))^^.^
Again, by equation X. article 21, we have
136 RELATIONS BETWEEN SEVERAL [BoOK 1.
and hence,
[9] '^=ie{0-l){c-l),
This equation 10 combined with 8 gives
/-\- r — (c -f- -) cos/, y/ r/
[11] «= '-^ •
Putting, therefore, in the same manner as in the ellipse
2 cos/ ' ' '
according as cos/ is positive or negative, we have
[12] « = -^^ '
— S (LA-iNc—J-f) cos f.\/r/
[12*] «= Y •
The computation of the quantity / or X is here made with the help of the auxil-
iary angle w in the same way as in the ellipse. Finally, we have from equation
XL article 22, (using the hyperbolic logarithms),
ii^^WCc— i-— - + ^) — logC'c+log-
= i.(C'+l)(.-l)-21og.,
or, C being eliminated by means of equation 8,
ht {c — \)cosf.slr/
\-h(cc ^
«i-
r^y^^ cc
21ogc.
In this equation we substitute for a its value from 12, 12* ; we then introduce
Sect. 3.] places in orbit. 13 7
the symbol m or M, with the same meaning that formulas 11, 11% article 88 give
it ; and finally, for the sake of brevity, we write
cc 4 log c
c c o
from which result the equations
[13] m={l—zf^{l — zfz,
[13*] iJf=_(X + .f+(XH-0)^^,
which involve only one unknown quantity, z, since Z is evidently a function of z
expressed by the following formula,
^_ (1 + 2 ^) V/(2^ + ^z) — log (y/ (1 -f g)-[- V^)
100.
In solving the equation 13 or 13* we will first consider, by itself, that case in
which the value of z is not great, so that Z can be expressed by a series proceed-
ing according to the powers of z and converging rapidly. Now we have
(i + 2^)v/(04-^^) = ^*4-f^*+l^*-..,
and so the numerator of ^ is | s^ -f- 1 0^ . . . ;
and the denominator, 2 0^ -[- 3 s^ . . . ,
whence,
^ A ^ 0 . . . .
In order to discover the law of progression, we differentiate the equation
2(0 + 00f ^=(1-1-20)^/(^4-00) — log (v/(l + 0) + v/0),
whence results, all the reductions being properly made,
2(0 + 00)*^-f-3^(l + 20)s/(0 + 00) = 4v/(0 + 00),
18
138 RELATIONS BETWEEN SEVERAL [BoOK 1.
or
(2. + 2..)^=4-(3 + 6.)Z,
whence, in the same manner as in article 90, is deduced
ry , 4.6 I 4.6.8 4.6.8.10 . , 4.6.8.10.12 .
^=^-375^ + 3-75:7""- 375777^^ + 3.5.7.9.11 " " ^*«-
It is evident, therefore, that Z depends upon — s in axactly the same manner
as X does upon x above in the ellipse ; wherefore, if we put
C also will be determined in the same manner bj — z as |, above, by a?, so that
we
have
[14]
L:
or,
1 +
t¥3^
l+etc.
c=
"1 +
1 + lf^
I + tVb^
1 + etc.
In this way the values of t are computed for z to single thousandths, from 2=0
up to s =: 0.3, which values are given in the third column of table HI.
101.
By introducing the quantity I, and putting
also
Vt rn mm J
[15] f+,+^ = !',or
[15T ^^ = ^,
Sect. 3.] places m orbit. 139
equations 13, 13* assume the form,
[16] ^^^^ = ^.
[16*] ^L+^f^^H,
and so, are wholly identical with those at which we arrived in the ellipse (15, 15*,
article 91). Hence, therefore, so far as ^ or ^ can be considered as known, t/ or
Y can be deduced, and afterwards we shall have
[17] ^=1--^,
[17*] ^ = f?-X.
From these we gather, that all the operations directed above for the elhpse serve
equally for the hyperbola, up to the period when t/ or Y shall have been deduced
from h or J3[; but after that, the quantity
mm , r ^^
yy ' ^^'
which, in the ellipse, should become positive, and in the parabola, 0, must in the
hyperbola become negative : the nature of the conic section will be defined by
this criterion. Our table will give t, from z thus found, hence will arise the cor-
rected value of h or H, with which the calculation is to be repeated until all
parts exactly agree.
After the true value of s is found, c might be derived from it by means of the
formula
(? = l-f 2^4-2^(0 + ^^),
but it is preferable, for subsequent uses, to introduce also the auxiliary angle n,
to be determined by the equation
tan2 w== 2\/(s-|-s0);
hence we have
(? = tan2 w 4- v/(l + tan2 2 «) = tan (45° + «).
140 RELATIONS BETWEEN SEVERAL [BoOK 1.
102.
Since y must necessarily be positive, as well in the hyperbola as in the ellipse,
the solution of equation 16 is, here also, free from ambiguity : -j- but with respect
to equation 16*, we must adopt a method of reasoning somewhat different from
that employed in the case of the eUipse. It is easily demonstrated, from the the-
ory of equations, that, for a positive value of 3%, this equation (if indeed it has
any positive real root) has, with one negative, two positive roots, which will either
both be equal, that is, equal to
i y/ 5 _ i = 0.20601,
or one will be greater, and the other less, than this Hmit. We demonstrate in
the following manner, that, in our problem (assuming that z is not a large
quantity, at least not greater than 0.3, that we may not abandon the use of the
third table) the greater root is always, of necessity, to be taken. K in equation
13* in place of Jf, is substituted Zy/(X-|-^),we have
Y^\ = {L^z)Z>{\-^z)Z, or
T7-»^ 1 ^ I 4. 6 4. 6. 8 o 1 ,
^> 3 -sTs ^ + 0:7^^-3X7:9^ +^*«-
whence it is readily inferred that, for such small values of z as we here suppose,
Y must always be > 0.20601. In fact, we find, on making the calculation, that
z must be equal to 0.79858 in order that {\-\-z)Z may become equal to this
limit : but we are far from wishing to extend our method to such great values of z.
103.
When z acquires a greater value, exceeding the limits of table m., the equa-
tions 13, 13* are always safely and conveniently solved by trial in their un-
changed form ; and, in fact, for reasons similar to those which we have explained
t It will hardly be necessary to remark, that our table II. can be used, in the hyperbola, as well as
in the ellipse, for the solution of this equation, as long as h does not exceed its limit.
X The quantity H evidently cannot become negative, unless C > i 5 but to such a value of t, would
correspond a value oiz greater than 2.684, thus, far exceeding the limits of this method.
Sect. 3.] places in orbit. 141
in article 94 for the ellipse. In such a case, it is admissible to suppose the
elements of the orbit, roughly at least, known : and then an approximate value
of n is immediately had by the formula
, o sin f J rr'
tan 2 w = — rr^ — r,.
which readily follows from equation 6, article 99. z also will be had from n by
the formula
2 cos 2 w cos 2 )^ '
and from the approximate value of z, that value will be deduced with a few
trials which exactly satisfies the equation 13, 13*. These equations can also be
exhibited in this form,
(J ^^'-^^^.9(J ^-'-^^ |Sl|-^^P-^°Stan (45° + :
^=-(^ + 0-^^) +2(^ + ^5^) I t^2s2^. ■
and thus, s beiag neglected, the true value of n can be deduced.
104.
It remains to determine the elements themselves from z, n, or c. Putting
ayj [ee — 1) = /?, we shall have from equation 6, article 99,
[18] /3==^^^^.
^ -i ^ tan2w
combining this formula with 12, 12* article 99, we derive,
[19] ^(..-l) = tanv' = f^,
[19*] tanv=_?f^,
whence the eccentricity is conveniently and accurately computed ; a will result
from (i and \J [ee — 1) by division, and p by multiplication, so that we have.
142 RELATIONS BETWEEN SEVERAL [BoOK I.
2 {I — z) cos/, y^r/ 2 m m cos f.\j rr' kktt
tan'^2n yy tannin 4 y j' rr' cos*/ tan'' 2 w
— 2(Z-f-z)cos/.v^r/ — 2MMcos/.^rr' kktt
ta.n^2n TTta.n'2n 4:TYrr^cosytaji^2n*
sin/, tan/, ^r/ yysmf.taxif.t^rr' /yr'/sia2f\^
^ 2{l — z) ~ ¥,i^ ~ \ kt )
— sin/.tan/.y/r/ — rFsin/. tan/, y/r/ /r?Vsin2/\2
~ 2(2+^) ~~ 2J^M ~\ Yt /*
The third and sixth expressions for p, which are wholly identical with the form-
ulas 18, 18*, article 95, show that what is there said concerning the meaning
of the quantities y, Y, holds good also for the hyperbola.
From the combination of the equations 6, 9, article 99, is derived
by introducing therefore \\) and w, and by putting (7=: tan (45° -f-iV), we have
[20] tan2i\^^'"7^"/".
>- -J sm/coszw
C being hence found, the values of the quantity expressed by u in article 21, will
be had for both places ; after that, we have by equation III., article 21,
Q—c
tan J V
tan ^ v'
Gc — 1
(Cc-l-l)tanii^'
or, by introducing for C, c, the angles N, n,
sin (iV — n)
[21] tan i V
cos {N-\- n) tan \ \p
[22] tan^.'=:-^^-mi^.
>- -■ cos (iv — n) tan ^ i//
Hence wUl be determined the true anomalies v,v', the difference of which com-
pared with 2/ will serve at once for proving the calculation.
Finally, the interval of time from the perihelion to the time corresponding to
the first place, is readily determined by formula XI., article 22, to be
4
& \
cos 2 iy^cos 2 n ^^i^- ^^^ tan (45° + «) / '
Sect. 3.] ' places in orbit. 143
and, in the same manner, tlie interval of time from the perihelion to the time cor-
responding to the second place,
/C"°1^7;l::Vf+"'-Vp.Iogtan H^^ + N) tan (46- + „)).
If, therefore, the first time is put = T — ^ t, and, therefore, the second = T-\- 1 1,
we have
[23] 2'=|'(^^-logtan(45° + i^)),
whence the time of perihelion passage will be known ; finally,
m ^ = T(^-Htan(45» + «)),
which equation, if it is thought proper, can be applied to the final proof of the
calculation.
105.
To illustrate these precepts, we will make an example from the two places
in articles 23, 24, 25, 46, computed for the same hyperbolic elements. Let,
accordingly,
v'—v = ^^°12' 0",or/ = 24° 6' 0", log r = 0.0333585, log / = 0.2008641,
^=51.49788 days.
Hence is found
w = 2° 45' 28".47, I = 0.05796039, •
^^ or the approximate value of ^ = 0.0644371 ; hence, by table 11.,
log 2/ 2/ = 0.0560848, "^ = 0.05047454, 0 = 0.00748585,
to which in table HI. corresponds t ^ 0.0000032. Hence the corrected value of
h is 0.06443691,
logj^y = 0.0560846, ^^'= 0.05047456, 0= 0.00748583,
°^ yy
which values require no further correction, because t, is not changed by them.
The computation of the elements is as follows : —
144
RELATIONS BETWEEN SEVERAL
[Book L
loo-g 7.8742399
0.0032389
log(l+.)
log2 . . .
8.9387394
0.3010300
log tan 2 n .
. . 9.2397694
2n=:
9°5rir.816
n =
4 55 35 .908
log sin/ .... 9.6110118
logs/r/ . . . . 0.1171063
Clog tan 2 w . . 0.7602306
log/3 0.4883487
log tan 1// . . . . 9.8862868
log a . . . . . 0.6020619
logp 0.3746355
(tliey should be 0.6020600 and 0.3746356)
log sin (iV — n)
C. log cos {J\f-\- n)
log cot ^ i/> . .
8.7406274
0.0112902
0.4681829
log tan iy . . . 9.2201005
hv= 9°25'29';97
v= 18 50 59.94
(it should be 18°51'0")
log 6 0.1010184
log tan 2 iV^ . . . 9.4621341
Clog cos 2 w . . 0.0064539
9.5696064
number = 0.37119863
hyp log tan (45° -f-iV) = 0.28591251
log tan/ 9.6506199
log ^ tan 2 w .... 8.9387394
C.log(/ — 0) . . . 1.2969275
log tan 1// 9.8862868
^= 37°34'59".77
(it should be 37'^ 35' 0")
Clog I sin/
log tan 2 to .
C log cos 2 w
log sin t// . .
log tan 2 N .
2N =
iV =
i\^ — w =
log sin(iV^-|- n]
Clogcos(iV—
0.6900182
8.9848318
0.0020156
9.7862685
n)
9.4621341
16° 9' 46^.253
8 4 53 .127
3 9 17 .219
13 0 29 .035
. 9.3523527
. 0.0006587
log cot it/; . . . . 0.4681829
log tan iy' .... 9.8211943
^v'= 33°3r29".93
v'= 67 2 59.86
(it should be 67° 3' 0")
loge 0.1010184
log tan 2 w . . . . 9.2397694
Clogcos2iV' . . . 0.0175142
9.3583020
number = 0.22819284
hyplogtan(45°4-w) = 0.17282621
Difference
0.08528612
Difference =
0.05536663
Sect. 3.] places in orbit. 145
log 8.9308783 log 8.7432480
|log« 0.9030928 flog a 0.9030928
C.iog^ 1.7644186 C.logy^ 1.7644186
logT 1.5983897 l^g^ 0.3010300
T= ' 39.66338 logt 1.7117894
t= 51.49788
Therefore, the perihelion passage is 13.91444 days distant from the time
corresponding to the first place, and 65.41232 days from the time corresponding
to the second place. Finally, we must attribute to the limited accuracy of the
tables, the small differences of the elements here obtained, from those, according
to which, the given places had been computed.
106.
In a treatise upon the most remarkable relations pertaining to the motion
of heavenly bodies in conic sections, we cannot pass over in silence the elegant
expression of the time by means of the major semiaxis, the sum r<-\- r, and the
chord joining the two places. This formula appears to have been first discovered,
for the parabola, by the illustrious Euler, (Miscell. Berolin, T. VII. p. 20,) who
nevertheless subsequently neglected it, and did not extend it to the ellipse and
hyperbola : they are mistaken, therefore, who attribute the formula to the illus-
trious Lambert, although the merit cannot be denied this geometer, of having
independently obtained this expression when buried in oblivion, and of having
extended it to the remaining conic sections. Although this subject is treated by
several geometers, stiU the careful reader will acknowledge that the following
explanation is not superfluous. We begin with the elliptic motion.
We observe, in the first place, that the angle 2/ described about the sun
(article 88, from which we take also the other symbols) may be assumed to be
less than 360° ; for it is evident that if this angle is increased by 360°, the time
is increased by one revolution, or
^^°=a^X 365.25 days.
19
146 RELATIONS BETWEEN SEVEEAL [BoOK 1.
Now, if we denote the chord by ^, we shall evidently have
()Q = {r cos v' — r cos vf -\- {r sin y' — r sin vf,
and, therefore, by equations Vm,, IX., article 8,
Q Q =1 a a {cos U' — cos^)^-f-^<^ cos^^) (sin^' — sin^f
:=! 4:aa sin^ (/ {sin? G -j- cos^ cp cos^ G) = 4zaa sin^ y (1 — ee cos^ G) .
We introduce the auxiliary angle h such, that cos h = e cos G ; at the same time,
that all ambiguity may be removed, we suppose h to be taken between 0° and
180°, whence sin h will be a positive quantity. Therefore, as y lies between the
same limits (for if 2 ^ should amount to 360° or more, the motion would attain to,
or would surpass an entire revolution about the sun), it readily follows from the
preceding equation that 9 = 2a sin^ sin^, if the chord is considered a positive
quantity. Since, moreover, we have
?--|-/==2a(l — ecos^cos(?) =: 2a{l — cos^cos A),
it is evident that, if we put h — ^ = (J , /« -|-^ = g , we have,
[1] r + /— ^ = 2a(l— cos(^) = 4«sinH^,
[2] r -\- r -\-Q^2a{l — cosf) = 4a sin^ ^ « .
Finally, we have
kt = a- {1g — 2 esin^ cos G) = a'- {2g — 2 sin^ cos^),
or
[3] Jet = a' {t — smE — {d — smd)).
Therefore, the angles d and e can be determined by equations 1, 2, from
r -|- /, q, and a ; wherefore, the time t will be determined, from the same equa-
tions, by equation 3. If it is preferred, this formula can be expressed thus :
7 7 1/ 2a—(r-\-r')—Q . 2a—(r4-r')—Q
/ct = a U arc cos ^rJ — ^ — ^ — sm arc cos ~ — - — ^
\ 2a 2a
2a—Cr-[-r')4-Q , • 2a— (r + r') + 0\
— arc cos \ '^ ^ ' '^ -4- sm arc cos v -r vry i _
2a ' 2a >
But an uncertainty remains in the determination of the angles (^, e, by their
cosines, which must be examined more closely. It appears at once, that d
must lie between — 180° and -j- 180°, and t between 0° and 360° : but thus
Sect. 3.] places in orbit. 147
both angles seem to admit of a double, and the resulting time, of a quadruple,
determination. "We have, however, from equation 5, article 88,
cos/. \J rr' =^a (cos_^ — cos ^) = 2 a sin I (^ sin ^ e :
now, sin i 6 is of necessity a positive quantity, whence we conclude, that cos/
and sin i d are necessarily affected by the same sign ; and, for this reason, that
d is to be taken between 0° and 180°, or between — 180° and 0° according as cos/
happens to be positive or negative, that is, according as the heliocentric motion
happens to be less or more than 180°. ' Moreover, it is evident that d must neces-
sarily be 0°, for 2/= 180°. In this manner d is completely determined. But
the determination of the angle e continues, of necessity, doubtful, so that two
values are obtained for the time, of which it is impossible to determine the true
one, mil ess it is known from some other source. Finally, the reason of this
phenomenon is readily seen : for it is known that, through two given points, it
is pos-sible to describe two different ellipses, both of which can have their focus
in the same given point and, at the same time, the same major semiaxis j* but
the motion from the first place to the second in these ellipses is manifestly per-
formed in unequal times.
107.
Denoting by x any arc whatever between — 180° and -J- 180°, and by s the
sine of the arc J ;f , it is known that,
Jx=^+|.^^« + i.y^^ + |.^^^+etc.
Moreover, we have
and thus
i sin X = s ^ {1 — ss) = s— i s^ —^s^ — ^^^ s^ — etc.
X-^X=HU' + i4^'-^hl^^^'+hl^,s^^etc.
* A circle being described from the first place, as a centre, witb tbe radius 2 a — r, and anoiher,
from the second place, with the radius 2 a — /, it is manifest that the other focus of the ellipse lies in the
intersection of these circles. Wherefore, since, generally speaking, two intersections are given, two dif-
ferent ellipses will be produced.
148 RELATIONS BETWEEN SEVERAL [BoOK I.
We substitute in this series for s, successively
3.
and we multiply tlie results hy a^ ; and thus obtain respectively, the series,
T^h2 ^s{r-\-/ — QY-{- etc.
T¥f32 ^(^ + ^'+Q)^4-etc.
the sums of which we will denote by T, U. Now it is easily seen, since
2sinH=+^^:+^,
the upper or lower sign having effect according as 2/ is less or more than 180°,
that
the sign being similarly determined. In the same manner, if for £ is taken the
smaller value, inferior to 180°, we have
«*(8 — sin e) = U;
but the other value, which is the complement of the former to 360°, being taken,
we evidently have
a^ (e — sin £) = a* 360° — J/:
Hence, therefore, are obtained two values for the time t,
UTT t a^360° U+T
-^,and-^ 1--
108.
If the parabola is regarded as an ellipse, of which the major axis is infinitely
great, the expression for the time, found in the preceding article, passes into
Jlii^ + r^ + QfTir + Z-gf):
Sect. 3.] places in orbit. 149
but since this derivation of the formula might perhaps seem open to some doubts,
we will give another not depending upon the ellipse.
Putting, for the sake of brevity,
tsiniv = &, tan ^ / = 6', we have r=^p{l-\-&6),r'=ip{l-\- &'&'),
l — dd , \ — e'e' . 26 . , 2 6'
— cos v' = , . .,.„ sm V ■=-—^-—, sm v' :
Hence follow
r' cos v' — rcosv=ip{&6 — ^'^'\ r' sin v' — r sin v =^p {&' — 6),
and thus
Now it is readily seen that ^' — &= — [^^-^ ^ , is a positive quantity : putting,
therefore,
V/(l + l(^'-f ^f)=7j, we have Q=p{&' — 6)rj.
Moreover,
r -}- r' = i p {2 -{- 66 -\- &' 6') =p (rjrj -^ i {&' — Bf) :
wherefore, we have
From the former equation is readily deduced,
as fj and 6' — & are positive quantities ; but since i {&' — 6) is smaller or greater
than rj, according as
rjrj _ 1 (^'_ ^)2 = 1 _|_ ^&'— 2^^/
cos -I y COS I- v'
is positive or negative, we must, evidently, conclude from the latter equation that
+ \f'±j=^ = V-H6'-6),
in which the upper or lower sign is to be adopted, according as the angle de-
scribed about the sun is less than 180°, or more than 180^
150 RELATIONS BETWEEN SEVERAL [BoOK I.
From the equation, wMch in article 98 follows the second equation, we have,
moreover,
whence readily follows,
the upper or lower sign taking effect, as 2/ is less or more than 180°.
109.
If, in the hyperbola, we take the symbols a, 0, c, with the same meaning as in
article 99, we have, from equations Ylll., IX., article 21,
/ cos y' — rcosv=^ — ^ (e jlC — jAa
/ sin / — r sin f =: i (c )l C-\- -^) ccsj [ee — 1) ;
and consequently,
Let us suppose that y is a quantity determined by the equation
since this is evidently satisfied by two values, the reciprocals of each other, we
may adopt the one which is greater than 1. In this manner
Moreover,
. + /=|«(.(. + l)(^+^)_4) = ia((. + i)(y + i)-4),
and thus,
Sect. 3.] places in orbit. 151
Putting, therefore,
we necessarily have
but in order to decide the question whether J ^- — y/- is equal to -|- 2 w or — 2n,
it is necessary to inquire whether y is greater or less than c : but it follows readily
from equation 8, article 99, that the former case occurs when 2/ is less than
180°, and the latter, when 2/ is more than 180°. Lastly, we have, from the same
article,
J=K^+,-)(^-J)-2l0g^=K^^-el)-*G--)-l0g., + l0g^
+ 21og(v/(l-[-72n) + w),
the lower signs belonging to the case of 2/> 180°. Now, log (v/(l -\- mm) -j- m)
is easily developed into the following series : —
This is readily obtained from
dlog (^\J{l-\-mm)-{-m)^=
Am
SJ {l-\-mmy
There follows, therefore, the formula
2mv/(l + mm) — 21og(v/(14-mm)+m) = 4(i;;i3__i^i^5_j_i^l:|^^7_etc.),
and, likewise, another precisely similar, if jw is changed to n. Hence, finally, if we
put
^=H'-+'-'-?)*-A.J(r + /-9)*+,/„.i(.+/-?)*
T"S'432
.1
[r-\-/ — qf-\-eiG.
— T^ls 2 • -^ (^ + ^' + 9)* + etc.
152 RELATIONS BETWEEN SEVERAL PLACES IN ORBIT. [BoOK I.
we obtain
which expressions entirely coincide with those given in article 107, if a is there
changed into — a.
Finally, these series, as well for the ellipse as the hyperbola, are eminently
suited to practical use, when a or a possesses a very great value, that is, where the
conic section resembles very nearly the parabola. In such a case, the methods
previously discussed (articles 85-105) might be employed for the solution of the
problem : but as, in our judgment, they do not furnish the brevity of the solution
given above, we do not dwell upon the further explanation of this method.
FOURTH SECTION.
EELATIONS BETWEEN SEVERAL PLACES IN SPACE.
110.
The relations to be considered in this section are independent of the nature of
the orbit, and will rest upon the single assumption, that all points of the orbit lie
in the same plane with the sun. But we have thought proper to touch here upon
some of the most simple only, and to reserve others more complicated and special
for another book.
The position of the plane of the orbit is fully determined by two places of
the heavenly body in space, provided these places do not lie in the same straight
line with the sun. Wherefore, since the place of a point in space can be assigned
in two ways, especially, two problems present themselves for solution.
We will, in the first place, suppose the two places to be given by means of
heliocentric longitudes and latitudes, to be denoted respectively by l, I', {^, (^ : the
distances from the sun will not enter into the calculation. Then if the longitude
of the ascending node is denoted by 9, , the inclination of the orbit to the ecliptic
by i, we shall have,
tan ^ = tan ^ sin [l — Q),
tan /3' = tan i sin (X' — U).
The determination of the unknown quantities Q , tan i, in this place, is referred
to the problem examined in article 78, 11. We have, therefore, according to the
first solution,
tan ^ sin (X — Q,) = tan ^ ,
tan/ cos (X — Q,)
tan ^' — tan (3 cos Qf — X)
20 (153)
154 RELATIONS BETWEEN SEVERAL [BoOK I.
likewise, accordiag to the third solution, we find Q by equation
and, somewhat more conveniently, if the angles ^, /5', are given immediately, and
not by the logarithms of their tangents : but, for determining i, recourse must be
had to one of the formulas
tan i = -T-
tan^
sin(A— g^) sin(?/— g^)'
Finally, the uncertainty in the determination of the angle
X — Q,or H+H' — Q,
by its tangent will be decided so that tan^ may become positive or negative,
according as the motion projected on the echptic is direct or retrograde : this
uncertainty, therefore, can be removed only in the case where it may be ap-
parent in what direction the heavenly body has moved in passing from the first
to the second place ; if this should be unknown, it would certainly be impossi-
ble to distinguish the ascending from the descending node.
After the angles 9,,i, are found, the arguments of the latitude it,u', will be
obtained by the formulas,
, tana— Q) X y tan(l'—Q,)
cos I ' cost '
which are to be taken in the first or second semicircle, according as the corre-
sponding latitudes are north or south. To these formulas we add the following,
one or the other of which can, at pleasure, be used for proving the calculation : —
cos u=cos§ cos {X — 9,), cos u' = cos /5' cos {X' — 9,),
sin/3 . / sins'
smw = -T-^., sm.u = ~-r,
^ I -' cose ^ ' COS I
Sect. 4.] places in space. I.5.5
111.
Let us suppose, in the second place, the two places to be given by means of
their distances from three planes, cutting each other at right angles in the sun ;
let us denote these distances, for the first place, by x, y, z, for the second, by
x', y', s, and let us suppose the third plane to be the ecliptic itself, also the posi-
tive poles of the first and second planes to be situated in N, and 90° -j- N. We
shall thus have by article 53, the two radii vectores being denoted by r, /,
x^rcos,u cos {N — Q,)-\-rmi.u sin (iV — 9> ) cos i, .
y = r sin M cos [N — Q> ) cos i — r cos u sin [N — 9,),
g = r sin % sin i
a/ = / cos u' cos [N — 9,)-\-r sin u' sin {N — 9, ) cos «',
1/ z=r' sin u' cos {N — 9, ) cos i — r' cos u' sin (iV — 9, ),
z' =. r' sin 1/ sin i.
Hence it follows that '
zy — ys =^ rr sin («' — u) sin [N — 9 ) sin i,
xz' — zx' =^ rr sin {id — u) cos [ISf — 9 ) sin i,
xy' — yx' = rr sin [u' — u) cos i.
From the combination of the first formula with the second will be obtained N — 9
and rr' sin [%{ — u) mii, hence and from the third formula, i and rr' sin {11 — u)
will be obtained.
Since the place to which the coordinates x', y', z', correspond, is supposed pos-
terior in time, 11! must be greater than u : if, moreover, it is known whether the
angle between the first and second place described about the sun is less or greater
than two right angles, rr Bm{ii — u)wii and rr'sin(««' — u) must be positive
quantities in the first case, negative in the second : then, accordingly, N — 9
is determined without doubt, and at the same time it is settled by the sign of
the quantity xy — yx, whether the motion is direct or retrograde. On the othei
hand, if the direction of the motion is known, it will be possible to decide from
the sign of the quantity xy' — y x, whether u' — u is to be taken less or greater
than 180°. But if the direction of the motion, and the nature of the angle
156 EELATIONS BETWEEN SE\T:RAL [Book I
c"! escribed about the sun are altogether unkno^vn, it is evident that we cannot dis-
tinguish between the ascending and descending node.
It is readily perceived that, just as cos i is the cosine of the inclination of
the plane of the orbit to the third plane, so sin (iV — Q, ) sin i, cos {N — Q> ) sin i,
are the cosines of the inclinations of the plane of the orbit to the first and second
j)lanes respectively ; also that r r' sin {%{ — ii) expresses the double area of the tri-
angle contained between the two radii vectores, and zy' — yz, xz — zx', x y' — yx\
the double area of the projections of this triangle upon each of the planes.
Lastly, it is evident, that any other plane can be the third plane, provided,
only, that all the dimensions defined by their relations to the ecliptic, are referred
to the third plane, whatever it may be.
112.
Let rr", y", z", be the coordinates of any third place, and v!' its argument of
the latitude, r" its radius vector. We will denote the quantities / r" sin [u" — u'),
rr"mi.{v!' — u),rr'wa.[u' — m), which are the double areas of the triangles be-
tween the second and third radii vectores, the first and third, the first and second,
respectively, by n, ri, n". Accordingly, we shall have for x", y", z'\ expressions
similar to those which we have given in the preceding article for x, y, z, and
x,y', z ; whence, with the assistance of lemma I, article 78, are easily derived the
following equations : —
{)=znx — nV -\- n"x",
0 = ny — nY -\- n"y",
0 =znz — 7i'z' -f- wV.
Let now the geocentric longitudes of the celestial body corresponding to these
three places be a, a', a"\ the geocentric latitudes, /^, \V , 1^"; the distances from the
earth projected on the ecliptic, (^, d', 8"; the corresponding heliocentric longitudes
of the earth, X, U, II' \ the latitudes, B, B', B" , which we do not put equal to
0, in order to take account of the parallax, and, if thought proper, to choose
any other plane, instead of the ecliptic ; lastly, let i>, Z>', D", be the distances of
the earth from the sun projected upon the echptic. If, then, x, y, z, are expressed
Sect. 4.] places in space. I57
by means of L, B, D, «, /?, d, and the coordinates relating to the second and third
places in a similar manner, the preceding equations will assume the following
form : —
[1] 0 = ?z (d~ cos a -|- Z> cos L) — n [d' cos a -\- D' cos L')
[2] 0=M((ysina4-i>sinX) — ;/(rsina'4-i)'sinX')
+ n" {d" sin a" + D" sin L"),
[3] 0 = w ((^ tan (5 + i? tan B) — n{d' tan §' + 1/ tan B')
+ n" {d" tan (i" + D" tan ^'').
If a, /i, i>, X, ^, and the analogous quantities for the two remaining places, are
here regarded as known, and the equations are divided by n', or by if, five un-
known quantities remain, of which, therefore, it is possible to eliminate two, or to
determine, in terms of any two, the remaining three. In this manner these three
equations pave the way to several most important conclusions, of which we will
proceed to develop those that are especially important.
113.
That we may not be too much oppressed with the length of the formulas, we
will use the following abbreviations. In the first place we denote the quantity
tan (i sin [a" — a) -(- tan (i' sin (a — a") -\- tan /3" sin {a —a)
by (0. 1. 2) : if, in this expression, the longitude and latitude corresponding to
any one of the three hehocentric places of the earth are substituted for the longi-
tude and latitude corresponding to any geocentric place, we change the number
answering to the latter in the symbol (0. 1. 2.) for the Roman numeral which
corresponds to the former. Thus, for example, the symbol (0. 1. 1.) expresses the
quantity
tan /3 sin {L' — a') -f tan /?' sin (a — L') -f tan B' mi{a' — a),
also the symbol (0. 0. 2), the following,
tan (i sm {a" — L)-\- tan B sin (a —a") -j- tan §" sin {L — a).
We change the symbol in the same way, if in the first expression any two heho-
158 RELATIONS BETWEEN SEVERAL [BoOK I.
centric longitudes and latitudes of the earth whatever, are substituted for two
geocentric. If two longitudes and latitudes entering into the same expression are
only interchanged with each other, the corresponding numbers should also be
interchanged ; but the value is not changed from this cause,' but it only becomes
negative from being positive, or positive from negative. Thus, for example, we
have
(0. 1. 2) = — (0. 2. 1) = (1. 2. 0) = — (1. 0. 2) = (2. 0. 1) = — (2. 1. 0).
All the quantities, therefore, originating in this way are reduced to the nineteen
following : —
(0.1.2)
(o.i.O), (o.i.i), (o.i.n.), (0.O.2), (0.1.2), (o.n.2), (0.1.2), (1.1.2), (n.1.2),
(0. 0. 1), (0. 0. II), (0. 1, n.), (1. 0. L), (1. 0. n.), (i. i ii), (2. 0. i), (2. 0. n.),
(2.1.11),
to which is to be added the twentieth (0. 1. II.).
Moreover, it is easily shown, that each of these expressions multiplied by the
product of the three cosines of the latitudes entering into them, becomes equal
to the sextuple volume of a pyramid, the vertex of which is in the sun, and the
base of which is the triangle formed between the three points of the celestial
sphere which correspond to the places entering into that expression, the radius
of the sphere being put equal to unity. When, therefore, these three places lie in
the same great circle, the value of the expression should become equal to 0 ; and
as this always occurs in three heliocentric places of the earth, when we do not
take account of the parallaxes and the latitudes arising from the perturbations of
the earth, that is, when we suppose the earth to be exactly in the plane of the
ecliptic, so we shall always have, on this assumption, (0. 1. II.) =^ 0, which is, in
fact, an identical equation if the ecliptic is taken for the third plane. And fur-
ther, when B, B', B", each, = 0, all those expressions, except the first, become
much more simple ; every one from the second to the tenth will be made up of
two parts, but from the eleventh to the twentieth they will consist of only one
term.
Sect. 4.] places in space. 159
114.
By multiplying equation [1] by sin a' tan B" — sin L" tan ^" , equation [2]
by cos U' tan 1^" — cos a" tan B" , equation [3] by sin {L" — a"\ and adding the
products, we get,
[4] 0 = ;z ((0. 2. n.) d -I- (0. 2. n.) i>) —i{ ((1. 2. n.) 8' + (I. 2. II.) Z>') ;
and in tbe same manner, or more conveniently by an interchange of the places,
simply
[5] 0 = « ((0. 1. 1.) 8 + (0. 1. 1.) X>) -h w" ((2. 1. 1.) 8" + (II. 1. 1.) i)")
[6] 0 = w' ((1. 0. 0.)^' + (I. 0. 0.)i>') — 1{' ((2. 0. 0.)r -j- (II. 0. 0.) i/').
If, therefore, the ratio of the quantities w, n, is given, with the aid of equation 4,
we can determine 8' from d, or d from d' ; and so likewise of the equations 5, 6.
From the combination of the equations 4, 5, 6, arises the following,
P^-, (0. 2. II.) a +(0.2. II.) i) (i.o.o.)y+(i.o.o.)D- (2.i.i.)y+(ii.i.i.)2y^__
L'-l (0.1.L)5-i-(O.l.I.)i> '^ (1.2.II.)5'-|-(I.2.II.)Z)'^ (2.0.O.)5"-f (IL0.O.)Zy'~ "^^
by means of which, from two distances of a heavenly body from the earth, the
third can be determined. But it can be shown that this equation, 7, becomes
identical, and therefore unfit for the determination of one distance from the other
two, when
B=B'=B"=0,
and
tan /?' tan /3'' sin (Z — a) sin {L" — L') -^ tan |S" tan (3 sin {L' — a') sin (X — X")
-f tan (^ tan (^ sin (X" — a") sin (X' — X) = 0.
The following formula, obtained easily from equations I, 2, 3, is free from this
inconvenience : —
[8] (0. 1. 2.) dd'r -f (0. 1. 2) Bd'r + (0. 1. 2) D'dd" + (0. 1. n.) ly'dd'
-I- (0. 1, n.) B'l/'d + (0. 1. n.) x>x>"d' -f- (o. i. 2) x>x>x + (O. i n.) bd'b" = o.
By multiplying equation 1 by sin a' tan (3'^ — sin a" tan j3', equation 2 by
cos a'' tan |3' — cos a' tan j3", equation 3 by sin («'' — a'), and adding the products,
we get
[9] Q = n({0.1.2) d -\- {0.1.2) B)— n' (1.1.2) jy^n'' (11.1.2) B"'
160 RELATIONS BETWEEN SEVERAL PLACES IN SPACE. [BoOK I,
and in the same manner,
[10] 0 = ?^(0.O.2.)Z> — «'((0.1.2)(^'-f (O.I.2)2X)-f n"(0.n.2)iX',
[11] 0 = n (0. 1. 0)D — n' (0. 1. 1.) 1/ + n" ((0. 1. 2) ^ + (0. 1. H.) !>').
By means of these equations the distances d, d', d", can be derived from the
ratio between the quantities n, n, n", when it is known. But this conclusion only
holds in general, and suffers an exception when (0.1.2)= 0. For it can be shown,
that in this case nothing follows from the equations 8, 9, 10, except a necessary
relation between the quantities n, n', n", and indeed the same relation from each
of the three. Analogous restrictions concerning the equations 4, 5, 6, will readily
suggest themselves to the reader.
Finally, all the results here developed, are of no utility when the plane of the
orbit coincides with the ecliptic. For if /?, §', /i", B, B B" are aU equal to 0,
equation 3 is identical, and also, therefore, all those which follow.
SECOND BOOK.
INVESTIGATION OF THE ORBITS OF HEAVENLY BODIES FROM GEOCENTRIC
OBSERVATIONS.
FIEST SECTION.
DETERMINATION OP AN ORBIT FROM THREE COMPLETE OBSERVATIONS.
115.
Seven elements are required for the complete determination of the motion
of a heavenly body in its orbit, the number of v^hich, however, may be dimin-
ished by one, if the mass of the heavenly body is either known or neglected ;
neglecting the mass can scarcely be avoided in the determination of an orbit
wholly unknown, where all the quantities of the order of the perturbations must
be omitted, until the masses on which they depend become otherwise known.
Wherefore, in the present inquiry, the mass of the body being neglected, we re-
duce the number of the elements to six, and, therefore, it is evident, that as many
quantities depending on the elements, but independent of each other, are re-
quired for the determination of the unknown orbit. These quantities are neces-
sarily the places of the heavenly body observed from the earth ; since each one
of which furnishes two data, that is, the longitude ajid latitude, or the right ascen-
sion and dechnation, it will certainly be the most simple to adopt three geocentric
places which will, in general, be sufficient for determining the six unknown ele-
ments. This problem is to be regarded as the most important in this work, and,
for this reason, vrill be treated with the greatest care in this section.
21 (161)
J62 DETERmNATION OF AN ORBIT FROM [BoOK II.
But in the special case, in which the plane of the orhit coincides with the
ecliptic, and thus both the heliocentric and geocentric latitudes, from their nature,
vanish, the three vanishing geocentric latitudes cannot any longer be considered
as three data independent of each other: then, therefore, this problem would
remain indeterminate, and the three geocentric places might be satisfied by an
infinite number of orbits. Accordingly, in such a case, four geocentric longitudes
must, necessarily, be given, in order that the four remaining unknown elements
(the inclination of the orbit and the longitude of the node being omitted) may be
determined. But although, from an indiscernible principle, it is not to be ex-
pected that such a case would ever actually present itself in nature, nevertheless,
it is easily imagined that the problem, which, in an orbit exactly coinciding with
the plane of the ecliptic, is absolutely indeterminate, must, on account of the
limited accuracy of the observations, remain nearly indeterminate in orbits very
little inclined to the ecliptic, where the very slightest errors of the observations
are sufficient altogether to confound the determination of the unknown quan-
tities. Wherefore, in order to examine this case, it will be necessary to select
six data : for which purpose we will show in section second, how to determine an
unknown orbit from four observations, of which two are complete, but the other
two incomplete, the latitudes or declinations being deficient.
Finally, as all our observations, on account of the imperfection of the instru-
ments and of the senses, are only approximations to the truth, an orbit based
only on the six absolutely necessary data may be still liable to considerable
errors. In order to duninish these as much as possible, and thus to reach the
greatest precision attainable, no other method will be given except to accumulate
the greatest number of the most perfect observations, and to adjust the elements,
not so as to satisfy this or that set of observations with absolute exactness, but
so as to agree with all in the best possible manner. For which pm^pose, we will
show in the third section how, according to the principles of the calculus of
probabilities, such an agreement may be obtained, as will be, if in no one place
perfect, yet in all the places the strictest possible.
The determination of orbits in this manner, therefore, so far as the heavenly
bodies move in them according to the laws of Kepler, will be carried to the
Sect. 1.] three complete observations. 163
highest degree of perfection that is desired. Then it will be proper to undertake
the final correction, in which the perturbations that the other planets cause in the
motion, will be taken account of: we will indicate briefly in the fourth section,
how these may be taken account of, so far at least, as it shall appear consistent
with our plan.
116.
Before the determination of any orbit from geocentric observations, if the
greatest accuracy is deshed, certain reductions must be applied to the latter on
account of nutation, precession, parallax, and aberration : these small quantities
may be neglected in the rougher calculation.
Observations of planets and comets are commonly given in apparent (that
is, referred to the apparent position of the equator) right ascensions and declina-
tions. Now as this position is variable on account of nutation and precession,
and, therefore, different for different observations, it will be expedient, first of all,
to introduce some fixed plane instead of the variable plane, for which purpose,
either the equator in its mean position for some epoch, or the ecliptic might be
selected : it is customary for the most part to use the latter plane, but the former
is recommended by some peculiar advantages which are not to be despised.
When, therefore, the plane of the equator is selected, the observations are in
the first place to be freed from nutation, and after that, the precession being
applied, they are to be reduced to some arbitrary epoch : this operation agrees
entirely with that by which, from the observed place of a fixed star, its mean
place is derived for a given epoch, and consequently does not need explanation
here. But if it is decided to adopt the plane of the ecliptic, there are two courses
which may be pursued : namely, either the longitudes and latitudes, by means of
the mean obliquity, can be deduced from the right ascensions and declinations
corrected for nutation and precession, whence the longitudes referred to the mean
equinox will be obtained ; or, the latitudes and longitudes will be computed more
conveniently from the apparent right ascensions and dechnations, using the appar-
ent obhquity, and will afterwards be freed from nutation and precession.
The places of the earth, corresponding to each of the observations, are com-
164 DETERMDWATION OF AN ORBIT FROM [BoOK 11.
puted from the solar tables, but they are evidently to be referred to the same
plane, to which the observations of the heavenly body are referred. For which
reason the nutation will be neglected in the computation of the longitude of the
sun ; but afterwards this longitude, the precession being applied, will be reduced
to the fixed epoch, and increased by 180 degrees; the opposite sign will be given
to the latitude of the sun, if, indeed, it seems worth while to take account of it :
thus will be obtained the heliocentric place of the earth, which, if the equator is
chosen for the fundamental plane, may be changed into right ascension and decli-
nation by making use of the mean obhquity.
117.
The position of the earth, computed in this manner from the tables, is the
place of the centre of the earth, but the observed place of the heavenly body
is referred to a point on the surface of the earth : there are three methods of
remedying this discrepancy. Either the observation can be reduced to the centre
of the earth, that is, freed from parallax ; or the heliocentric place of the earth
may be reduced to the place of observation, which is done by applying the
parallax properly to the place of the sun computed from the tables ; or, finally,
both positions can be transferred to some third point, which is most conveniently
taken in the intersection of the visual ray with the plane of the ecliptic ; the
observation itself then remains unchanged, and we have explained, in article 72,
the reduction of the place of the earth to this point. The first method cannot be
applied, except the distance of the heavenly body from the earth be approxi-
mately, at least, known : but then it is very convenient, especially when the
observation has been made in the meridian, in which case the declination only is
affected by parallax. Moreover, it will be better to apply this reduction imme-
diately to the observed place, before the transformations of the preceding article
are undertaken. But if the distance from the earth is still whoUy unknown,
recourse must be had to the second or third method, and the former will be em-
ployed when the equator is taken for the fundamental plane, but the third will
have the preference when all the positions are- referred to the ecliptic.
Sect. 1.] theee coiviplete observations. 165
118.
If the distance of a heavenly body from the earth answering to any observa-
tion is already approximately known, it may be freed from the effect of aberra-
tion in several ways, depending on the different methods given in article 71.
Let ^ be the true time of observation ; 6 the interval of time in which light passes
from the heavenly body to the earth, which results from multiplying 493' into the
distance ; I the observed place, I the same place reduced to the time ^ -)- ^ by
means of the diurnal geocentric motion ; V the place I freed from that part of the
aberration which is common to the planets and fixed stars ; L the true place of
the earth corresponding to the time t (that is, the tabular place increased by
20".25) ; lastly, 'X the true place of the earth corresponding to the time t — ^.
These things being premised, we shall have
I. I the true place of the heavenly body seen from 'L at the time t — ^.
n. / the true place of the heavenly body seen from L at the time t.
ni. V the true place of the heavenly body seen from L at the time t — ^.
By method L, therefore, the observed place is preserved unchanged, but the fic-
titious time t — ^ is substituted for the true, the place of the earth being com-
puted for the former ; method 11., applies the change to the observation alone, but
it requires, together with the distance, the diurnal motion ; in method III., the
observation undergoes a correction, not depending on the distance ; the fictitious
time t — ^ is substituted for the true, but the place of the earth corresponding to
the true time is retained. Of these methods, the first is much the most conven-
ient, whenever the distance is known well enough to enable us to compute the
reduction of the time with sufficient accuracy. But if the distance is whoUy un-
known, neither of these methods can be immediately appUed : in the first, to be
sure, the geocentric place of the heavenly body is known, but the time and the
position of the earth are wanting, both depending on the unknown distance ; in
the second, on the other hand, the latter are given, and the former is wanting;
finally, in the third, the geocentric place of the heavenly body and the position
of the earth are given, but the time to be used with these is wanting.
166 DETERmNATION OF AN ORBIT FROM [BoOK II.
"WTiat, therefore, is to be done with our problem, if, in such a case, a sohition
exact with respect to aberration is required? The simplest course undoubtedly
is, to determine the orbit neglecting at first the aberration, the effect of which can
never be important ; the distances will thence be obtained with at least such pre-
cision that the observations can be freed from aberration by some one of the
methods just explained, and the determination of the orbit can be repeated with
greater accuracy. Now, in this case the third method will be far preferable to the
others : for, in the first method all the computations depending on the position of
the earth must be commenced again from the very beginning; in the second (which
in fact is never applicable, unless the number of observations is sufficient to obtain
from them the diurnal motion), it is necessary to begin anew all the computations
depending upon the geocentric place of the heavenly body ; in the third, on the
contrary, (if the first calculation had been already based on geocentric places
freed from the aberration of the fixed stars) all the preliminary computations
depending upon the position of the earth and the geocentric place of the heavenly
body, can be retained unchanged in the new computation. But in this way it
will even be possible to include the aberration directly in the first calculation, if
the method used for the determination of the orbit has been so arranged, that
the values of the distances are obtained before it shall have been necessary to
introduce into the computation the corrected times. Then the double compu-
tation on account of the aberration will not be necessary, as will appear more
clearly in the further treatment of our problem.
119.
It would not be difficult, from the connection between the data and unknown
quantities of our problem, to reduce its statement to six equations, or even to less,
since one or another of the unknown quantities might, conveniently enough, be
eliminated : but since this connection is most complicated, these equations would
become very intractable ; such a separation of the unknown quantities as finally
to produce an equation containing only one, can, generally speaking, be regarded
Sect. 1.] three complete observations. 167
as impossible,* and, therefore, still less will it be possible to obtain a complete
solution of the problem bj direct processes alone.
But our problem may at least be reduced, and that too in various ways, to the
solution of two equations X = 0, F= 0, in which only two unknown quantities
X, y, remain. It is by no means necessary that x, y, should be two of the ele-
ments : they may be quantities connected with the elements in any manner
whatever, if, only, the elements can be conveniently deduced from them when
found. Moreover, it is evidently not requisite that X, Y, be expressed in explicit
functions oi x,y : it is sufficient if they are connected with them by a system of
equations»in such manner that we can proceed from given values of x, y, to the
corresponding values of X, Y.
120.
Since, therefore, the nature of the problem does not allow of a further reduc-
tion than to two equations, embracing indiscriminately two unknown quantities,
the principal point will consist, first, in the suitable selection of these unknown
quantities and arrangemeni of the equations, so that both X and Y may depend
in the simplest manner upon x, y, and that the elements themselves may follow
most conveniently from the values of the former when known : and then, it will
be a subject for careful consideration, how values of the unknown quantities satis-
fying the equations may be obtained by processes not too laborious. If this should
be practicable only by blind trials, as it were, very great and indeed almost intol-
erable labor would be required, such as astronomers who have determined the
orbits of comets by what is called the indirect method have, nevertheless, often
undertaken : at any rate, the labor in such a case is very greatly lessened, if, in
the first trials, rougher calculations suffice until approximate values of the im-
known quantities are found. But as soon as an approximate determination is
made, the solution of the problem can be completed by safe and easy methods,
which, before we proceed further, it will be well to explain in this place.
* When the observations are so near to each other, that the intervals of the times may be treated as
infinitely small quantities, a separation of this kind is obtained, and the whole problem is reduced to the
solution of an algebraic equation of the seventh or eighth degree.
168 DETERINIINATION Ol' AN ORBIT FROM [BoOK 11.
The equations X=0, F=i: 0 will be exactly satisfied if for x and // their
true values are taken ; if, on the contrary, values different from the true ones are
substituted for x and y, then X and Y will acquire values differing from 0. The
more nearly x and ?/ approach their true values, the smaller should be the result-
ing values of X and Y, and when their differences from the true values are very
small, it will be admissible to assume that the variations in the values of X and Y
are nearly proportional to the variation of x, if 2/ is not changed, or to the varia-
tion of y, if a; is not changed. Accordingly, if the true values of x and ?/ are
denoted by ^, r], the values of X and Y corresponding to the assumption that
x=^'^ -\-l, i/ = r}-\- fi, will be expressed in the form
X=al-\-(S^, Y=yl-\-dfj.,
in which the coefficients a, /5, y, d can be regarded as constant^ as long as I and ix
remain very small. Hence we conclude that, if for three systems of values of
X, t/, differing but little from the true values, corresponding values of X, Y have
been determined, it will be possible to obtain from them correct values of x, y so
far, at least, as the above assumption is admissible. Let us suppose that,
for ;?• = a, ^ = ^ we have X-= A, Y=:B,
x = d,y=y X= A Y= B',
x=d',y=l" X=A' Y=B",
and we shall have
A = a{a-l)^(i{h — ri), B = y {^ -I) -^H^ -^l
A:=.a{a'-l)J^(i{h' — r^),B' = r{a' — l)^-d{h' — ri),
A'=a{a"-^)Jr(i{h"-ri),B"=.y{d'-l)^d{h"-n).
From these we obtain, by eliminating a, §, y, d,
t _ g {A'B' — J!'B') -|- a' {A"B— A B") -]- d' (^ F —A'B)
^~ A'B' — A!'B-\-A!'B—AB'-\-AB — A'B '
_ h{A'B' — A!'B) + h' (A"B— A B') + h" (AB — A'B)
^~" A'B' — A"B-j-A"B—AB'-\-AB — A'B '
or, in a form more convenient for computation,
^=a-
(a^— a) (ATB — A B") -^{d' — a){A B — A'B)
A'B'~A"B-\-A"B—AB'-\-AB — A'B =
_^ I {h' — b)(A"B—AB')-{-{b" — b)(AB — A'B)
V—^'T A'B' — A"B-{-A"B—AB'-\-AB — A'B '
Sect. 1.] three complete observations. 1G9
It is evidently admissible, also, to interchange in these formulas the quantities
a, b, A, B, with a', b', A', B', or with a', V\ A", B".
The common denominator of all these expressions, which may be put under
the form {A' — A) [B" — B) — [A' — A) {B' — B), becomes
{^ad—iiy) {{a' — a) {h"—h) — {a" — a) {h' — b)) :
whence it appears that a, a, d', b, b', b" must be so taken as not to make
h"—h V—W
otherwise, this method would not be applicable, but would furnish, for the values
of I and 7], fractions of which the numerators and denominators would vanish at
the same time. It is evident also that, if it should happen that ad — (:?/ = 0, the
same defect wholly destroys the use of the method, in whatever way a, a', d',
h, b', b", may be taken. In such a case it would be necessary to assume for the
values of X the form
and a similar one for the values of Y, which being done, analysis would supply
Boiethods, analogous to the preceding, of obtaining from values of X, Y, computed
for four systems of values of x, y, true values of the latter. But the computation
in this way would be very troublesome, and, moreover, it can be shown that, in
such a case, the determination of the orbit does not, from the nature of the ques-
tion, admit of the requisite precision : as this disadvantage can only be avoided
by the introduction of new and more suitable observations, we do not here dwell
upon the subject.
121.
When, therefore, the approximate values of the unknown quantities are ob-
tained, the true values can be derived from them, in the manner just now ex-
plained, with all the accuracy that is needed. First, that is, the values of X, Y,
corresponding to the approximate values (a, h) will be computed : if they do not
vanish for these, the calculation wiU be repeated with two other values [a, b')
differing but little from the former, and afterwards with a third system [a", b")
22
170 DETEEMINATION OF AN ORBIT FROM [BoOK II.
unless X, Y, have vanislied for the second. Then, the true values ■will be de-
duced by means of the formulas of the preceding article, so far as the assumption
on which these formulas are based, does not differ sensibly from the truth. In
order that we may be better able to judge of which, the calculation of the values
of X, Y, will be repeated with those corrected values ; if this calculation shows
that the equations X= 0, Y:= 0, are, still, not satisfied, at least much smaller
values of X, Y, will result therefrom, than from the three former h3^otheses, and
therefore, the elements of the orbit resulting from them, will be much more exact
than those which correspond to the first hypotheses. If we are not satisfied
with these, it will be best, omitting that hypothesis which produced the greatest
differences, to combine the other two with a fourth, and thus, by the process of
the preceding article, to obtain a fifth system of the values of :r, ^ ; in the same
manner, if it shall appear worth while, we may proceed to a sixth hypothesis,
and so on, until the equations X=^0, Y=: 0, shall be satisfied as exactly as the
logarithmic and trigonometrical tables permit. But it will very rarely be neces-
sary to proceed beyond the fourth system, unless the first hypotheses were very
far from the truth.
122.
As the values of the unknown quantities to be assumed in the second and third
hypotheses are, to a certain extent, arbitrary, provided, only, they do not differ
too much from the first hypothesis ; and, moreover, as care is to be taken that the
ratio [a" — a) : {b" — b) does not tend to an equality with {a — a) : {b' — ^), it is
customary to put a'=a, b'^=b. A double advantage is derived from this; for, not
only do the formulas for ^, rj, become a little more simple, but, also, a part of the
first calculation will remain the same in the second hypothesis, and another part
in the third.
Nevertheless, there is a case in which other reasons suggest a departure from
this custom : for let us suppose X to have the form X' — x, and Y the form
F' — y, and the functions X', Y', to become such, by the nature of the problem,
that they are very little affected by small errors in the values of x, y, or that
\"d^/' Viy/' \d^/' \^)
Sect. 1.] three complete observations. 171
may be very small quantities, and it is evident tliat tlie differences between tlie
values of those functions corresponding to the system x=^, y ^=-% and those
which result from x = «, y = h, can be referred to a somewhat higher order
than the differences \ — a,ri — I; but the former values are X' = |, F' := i;, and
the latter X' =^a^A, Y' = b-\-B, wdience it follows, that a -{-A, h-\-B, are
much more exact values of x, y, than a, b. If the second hypothesis is based
upon these, the equations X= 0, F= 0, are very frequently so exactly satisfied,
that it is not necessary to proceed any further ; but if not so, the third hypoth-
esis will be formed in the same manner from the second, by making
it" = a'-\-A!^a^A^A:,b" = b'^B' = h-^B^B',
whence finally, if it is still not found sufficiently accurate, the fourth will be ob-
tained according to the precept of article 120.
123.
We have supposed in what goes before, that the approximate values of the
unknown quantities x,y, are already had in some way. Where, indeed, the
approximate dimensions of the whole orbit are known (deduced perhaps from
other observations by means of previous calculations, and now to be corrected by
new ones), that condition can be satisfied without difficulty, whatever meaning we
may assign to the unknown quantities. On the other hand, it is by no means a
matter of indifference, in the determination of an orbit still wholly unknown,
(which is by far the most difficult problem,) what unknown quantities we may
use ; but they should be judiciously selected in such a way, that the approximate
values may be derived from the nature of the problem itself Which can be done
most satisfactorily, when the three observations applied to the investigation of
an orbit do not embrace too great a heliocentric motion of the heavenly body.
Observations of this kind, therefore, are always to be used for the first determina-
tion, which may be corrected afterwards, at pleasure, by means of observations
more remote from each other. For it is readily perceived that the nearer the ob-
servations employed are to each other, the more is the calculation affected by their
unavoidable errors. Hence it is inferred, that the observations for the first de-
1Y2 DETEEMINATTON OF JlS ORBIT FEOM [BoOK 11.
termination are not to be picked out at random, but care is to be taken, ^rst, that
tliey be not too near each other, but then, also, that they be not too distant from
each other ; for in the first case, the calculation of elements satisfying the obser-
vations would certainly be most expeditiously performed, but the elements them-
selves would be entitled to little confidence, and might be so erroneous that they
could not even be used as an approximation : in the other case, we should aban-
don the artifices which are to be made use of for an approximate determination
of the unknown quantities, nor could we thence obtain any other determination,
except one of the rudest kind, or wholly insufficient, without many more hj-poth-
eses, or the most tedious trials. But how to form a correct judgment concerning
these limits of the method is better learned by frequent practice than by rules :
the examples to be given below will show, that elements possessing great accu-
racy can be derived from observations of Juno, separated from each other only 22
days, and embracing a heliocentric motion of 7° 35'; and again, that our method
can also be applied, with the most perfect success, to observations of Ceres, which
are 260 days apart, and include a heliocentric motion of 62° 55'; and can give,
with the use of four hypotheses or, rather, successive approximations, elements
agreeing excellently well with the observations.
124.
We proceed now to the enumeration of the most suitable methods based upon
the preceding principles, the chief parts of which have, indeed, already been ex-
plained in the first book, and require here only to be adapted to our purpose.
The most simple method appears to be, to take for x, ?/, the distances of the
heavenly body from the earth in the two observations, or rather the logarithms
of these distances, or the logarithms of the distances projected upon the ecliptic
or equator. Hence, by article 64, V., will be derived the hehocentric places and
the distances from the sun pertaining to those places ; hence, again, by article 110,
the position of the plane of the orbit and the heliocentric longitudes in it ; and
from these, the radii vectores, and the corresponding times, according to the prob-
lem treated at length in articles 85-105, all the remaining elements, by which,
it is evident, these observations will be exactly represented, whatever values may
Sect. 1.] three complete observations. 173
have been assigned to x, y. If, accordingly, tlie geocentric place for the time of
the third observation is computed by means of these elements, its agreement or
disagreement with the observed place will determine whether the assumed values
are the true ones, or whether they differ from them ; whence, as a double com-
parison will be obtained, one difference (in longitude or right ascension) can be
taken for X, and the other (in latitude or declination) for T. Unless, therefore,
the values of these differences come out at once = 0, the true values of x, y, may
be got by the method given in 120 and the following articles. For the rest, it is
in itself arbitrary from which of the three observations we set out : still, it is bet-
ter, in general, to choose the first and last, the special case of which we shall speak
directly, being excepted.
This method is preferable to most of those to be explained hereafter, on this
account, that it admits of the most general application. The case must be ex-
cepted, in which the two extreme observations embrace a heliocentric motion of
180, or 360, or 640, etc., degrees; for then the position of the plane of the orbit
cannot be determined, (article 110). It will be equally inconvenient to apply the
method, when the heliocentric motion between the two extreme observations
differs very little from 180° or 360°, etc., because an accurate determination of
the position of the orbit cannot be obtained in this case, or rather, because the
slightest changes in the assumed values of the unknown quantities would cause
such great variations in the position of the orbit, and, therefore, in the values of
-X", Y, that the variations of the latter could no longer be regarded as propor-
tional to those of the former. But the proper remedy is at hand ; which is, that
we should not, in such an event, start from the two extreme observations, but from
the first and middle, or from the middle and last, and, therefore, should take for
X, F, the differences between calculation and observation in the third or first
place. But, if both the second place should be distant from the first, and the
third from the second nearly 180 degrees, the disadvantage could not be removed
in this way ; but it is better not to make use, in the computation of the elements,
of observations of this sort, from which, by the nature of the case, it is wholly
impossible to obtain an accurate determination of the position of the orbit.
Moreover, this method derives value from the fact, that by it the amount of
174 DETEKMINATION OF A^ ORBIT FROM [BoOK II.
the variations which the elements experience, if the middle place changes while
the extreme places remain fixed, can be estimated without difficulty: in this way,
therefore, some judgment may be formed as to the degree of precision to be
attributed to the elements found.
125.
We shall derive the second from the preceding method by applying a slight
change. Starting from the distances in two observations, we shall determine aU
the elements in the same manner as before; we shall not, however, compute
from these the geocentric place for the third observation, but will only proceed
as far as the heliocentric place in the orbit ; on the other hand we wiU obtain the
same hehocentric place, by means of the problem treated in articles 74, 75, from
the observed geocentric place and the position of the plane of the orbit; these
two determinations, different from each other (unless, perchance, the true values
of X, y, should be the assumed ones), will furnish us X and Y, the difference be-
tween the two values of the longitude in orbit being taken for X, and the differ-
ence between the two values of the radius vector, or rather its logarithm, for Y.
This method is subject to the same cautions we have touched upon in the pre-
ceding article : another is to be added, namely, that the heliocentric place in orbit
cannot be deduced from the geocentric place, when the place of the earth happens
to be in either of the nodes of the orbit ; when that is the case, accordingly, this
method cannot -be applied. But it wUl also be proper to avoid the use of this
method in the case where the place of the earth is very near either of the nodes,
since the assumption that, to small variations of t, y, correspond proportional
variations of X, Y, would be too much in error, for a reason similar to that which
we have mentioned in the preceding article. But here, also, may be a remedy
sought in the interchange of the mean place with one of the extremes, to which
may correspond a place of the earth more remote from the nodes, except, per-
chance, the earth, in all three of the observations, should be in the vicinity of the
nodes.
Sect. 1.] three cor^iPLETE observations. 175
126.
The preceding metliod prepares the way directly for the iliird. In the same
manner as before, by means of the distances of the heavenly body from the earth
in the extreme observations, the corresponding longitudes in orbit together with
the radii vectores may be determined. With the position of the plane of the
orbit, which this calculation will have furnished, the longitude in orbit and the
radius vector will be got from the middle observation. The remaining elements
may be computed from these three heliocentric places, by the problem treated in
articles 82, 83, which process will be independent of the times of the observa-
tions. In this way, three mean anomalies and the diurnal motion will be known,
whence may be computed the intervals of the times between the first and second,
and between the second and third observations. The differences between these
and the true intervals will be taken for X and Y.
This method is less advantageous when the heliocentric motion includes a
small arc only. For in such a case this determination of the orbit (as we have
already shown in article 82) depends on quantities of the third order, and does
not, therefore, admit of sufficient exactness. The slightest changes in the values
of x,y, might cause very great changes in the elements and, therefore, in the val-
ues of -X, y, also, nor would it be allowable to suppose the latter proportional to
the former. But when the three places embrace a considerable heliocentric mo-
tion, the use of the method will undoubtedly succeed best, unless, indeed, it is
thrown into confusion by the exceptions explained in the preceding articles,
which are evidently in this method too, to be taken into consideration.
127.
After the three heliocentric places have been obtained in the way we have
described in the preceding article, we can go forward in the following manner.
The remaining elements may be determined by the problem treated in articles
85-105, first, from the first and second places with the corresponding interval of
time, and, afterwards, in the same manner, from the second and third places and
176 DETERMINATION OF AN ORBIT FROM [BoOK 11.
the corresponding interval of time : thus two values will result for each of the
elements, and from their differences any two may be taken at pleasure for X and
Y. One advantage, not to be rejected, gives great value to this method ; it is,
that in the first hypotheses the remaining elements, besides the two which are
chosen for fixing X and Y, can be entirely neglected, and will finally be deter-
mined in the last calculation based on the corrected values of x, y^ either from
the first combination alone, or from the second, or, which is generally preferable,
from the combination of the first place with the third. The choice of those two
elements, which is, commonly speaking, arbitrary, furnishes a great variet}^ of
solutions ; the logarithm of the semi-parameter, together with the logarithm of
the semi-axis major, may be adopted, for example, or the former with the eccen-
tricity, or the latter with the same, or the longitude of the perihelion with any
one of these elements: any one of these four elements might also be combined
with the eccentric anomaly corresponding to the middle place in either calcula-
tion, if an elliptical orbit should result, when the formulas 27-30 of article 96,
will supply the most expeditious computation. But in special cases this choice
demands some consideration ; thus, for example, in orbits resembling the parabola,
the semi-axis ma'or or its logarithm would be less suitable, inasmuch as excessive
variations of these quantities could not be regarded as proportional to changes of
x,y: in such a case it would be more advantageous to select - . But we give less
time to these precautions, because the fifth method, to be explained in the follow-
ing article, is to be preferred, in almost all cases, to the four thus far explained.
128.
.Let us denote three radii vectores, obtained in the same manner as in articles
125, 126, by r, r, r" ; the angular heliocentric motion in orbit from the second to
the third place by 2/, from the first to the third by If, from the first to the
second by 2/", so that we have
next, let
r'/'sin2/=w, r r" sin 2/' = ?2', r / sin 2/" =: w'' ;
Sect. 1.] three complete observations. 177
lastly, let the product of the constant quantity li (article 2) into the intervals of
the time from the second observation to the third, from the first to the third, and
from the first to the second be respectively, ^, ^' ^" . The double computation of
the elements is begun, just as in the preceding article, both from rr f" and ^",
and from //',/, ^ : but neither computation will be continued to the determina-
tion of the elements, but v^^ill stop as soon as that quantity has been obtained
which expresses the ratio of the elliptical sector to the triangle, and which is de-
noted above (article 91) by y or — Y. Let the value of this quantity be, in the
first calculation, -»/', in the second, -»]. Accordingly, by means of formula 18, arti-
cle 95, we shall have for the semi-parameter j>) the two values: —
But we have, besides, by article 82, a third value,
4 rr'i" sin/sin/' sin/"
V— n — n'^ri' '
which three values would evidently be identical if true values could have been
taken in the beginning for x and y. For which reason we should have
, , „ 4 /9(?"rrV' sin/sin/' sin/" n' dd"
' Tir{ nn' 2 7]7jrrr cosj cosj cosf
Unless, therefore, these equations are fully satisfied in the first calculation, we
can put
A = logV^>
Y=^n — n -\-n — ^r—j, — rj, ? — j-, t^-
' z riri r r r cos J cos/ cos/
This method admits of an application equally general with the second ex-
plained in article 126, but it is a great advantage, that in this fifth method the
first hypotheses do not require the determination of the elements themselves, but
stop, as it were, half way. It appears, also, that in this process we find that, as it
can be foreseen that the new hypothesis will not differ sensibly from the truth, it
will be sufiicient to determine the elements either from r, /,/", &", alone, or from
r', r",f, ^, or, which is better, from r, r" f, &'.
23
178 DETERMINATION OF AN ORBIT FROM [BoOK II.
129.
The five methods thus far explained lead, at once, to as many others which
differ from the former only in this, that the inclination of the orbit and the lon-
gitude of the ascending node, instead of the distances from the earth, are taken
for X and y. The new methods are, then, as follows : —
I. From X and j/, and the two extreme geocentric places, according to articles
74, 75, the heliocentric longitudes in orbit and the radii vectores are determmed,
and, from these and the corresponding times, all the remaining elements ; from
these, finally, the geocentric place for the time of the middle observation, the
differences of which from the observed place in longitude and latitude will fur-
nish X and Y.
The four remaining methods agree in this, that all three of the heliocentric
longitudes in orbit and the corresponding radii vectores are computed from the
position of the plane of the orbit and the geocentric places. But afterwards : —
II. The remaining elements are determined from the two extreme places only
and the corresponding times ; with these elements the longitude in orbit and
radius vector are computed for the time of the middle observation, the differences
of which quantities from the values before found, that is, deduced from the geo-
centric place, will produce X and Y:
in. Or, the remaining dimensions of the orbit are derived from all three
heliocentric places (articles 82, 83,) into which calculation the times do not enter :
then the intervals of the times are deduced, which, in an orbit thus found, should
have elapsed between the first and second observation, and between this last
and the third, and their differences from the true intervals will furnish us with
X and Y:
IV. The remaining elements are computed in two ways, that is, both by the
combination of the first place with the second, and by the combination of the
second with the third, the corresponding intervals of the times being used. These
two systems of elements being compared with each other, any two of the differ-
ences may be taken for X and Y:
V. Or lastly, the same double calculation is only continued to the values of
Sect. 1.] three complete observations. 179
the quantity denoted by y, in article 91, and then the expressions given in the
preceding article for X and Y, are adopted.
In order that the last four methods may be safely used, the places of the earth
for all three of the observations must not be very near the node of the orbit : on
the other hand, the use of the first method only requhes, that this condition may
exist in the two extreme observations, or rather, (since the middle place may be
substituted for either of the extremes,) that, of the three places of the earth,
not more than one shall he in the vicinity of the nodes.
130.
The ten methods explained from article 124 forwards, rest upon the assump-
tion that approximate values of the distances of the heavenly body from the
earth, or of the position of the plane of the orbit, are already known. When
the problem is, to correct, by means of observations more remote from each other,
the dimensions of an orbit, the approximate values of which are already, by
some means, known, as, for instance, by a previous calculation based on other
observations, this assumption will evidently be liable to no difficulty. But it does
not as yet appear from this, how the first calculation is to be entered upon when
all the dimensions of the orbit are still wholly unknown : this case of our problem
is by far the most important and the most difficult, as may be imagined from
the analogous problem in the theory of comets, which, as is well known, has
perplexed geometers for a long time, and has given rise to many fruitless
attempts. In order that our problem may be considered as correctly solved, that
is, if the solution be given in accordance with what has been explamed in the
119th and subsequent articles, it is evidently requisite to satisfy the folio win,:
conditions : — First, the quantities x, y, are to be chosen in such a manner, that
we can find approximate values of them from the very nature of the problem, at
all events, as long as the heliocentric motion of the heavenly body between the
observations is not too great. Secondly, it is necessary that, for small changes in
the quantities x, y, there be not too great corresponding changes in the quantities
to be derived from them, lest the errors accidentally introduced in the assumed
values of the former, prevent the latter from being considered as approximate.
180 DETERMINATION OF AN ORBIT FROM [BoOK II.
Thirdly and lastly, we require that the processes by which we pass from the quan-
tities X, y, to X, Y, successively, be not too complicated.
These conditions will furnish the criterion by which to judge of the excellence
of any method : this will show itself more plainly by frequent applications. The
method which we are now prepared to explain, and which, in a measure, is to be
regarded as the most important part of this work, satisfies these conditions so that
it seems to leave nothing further to be desired. Before entering upon the ex-
planation of this in the form most suited to practice, we will premise certain pre-
liminary considerations, and we will illustrate and open, as it were, the way to it,
which might, perhaps, otherwise, seem more obscure and less obvious.
131.
It is shown in article 114, that if the ratio between the quantities denoted
there, and in article 128 by w, w', rl', were known, the distances of the heavenly
body from the earth could be determined by means of very simple formulas.
Now, therefore, if
should be taken for ar, y^
(the symbols ^, ^\ ^", being taken in the same signification as in article 128) im-
mediately present themselves as approximate values of these quantities in that
case where the heliocentric motion between the observations is not very great :
hence, accordingly, seems to flow an obvious solution of our problem, if two dis-
tances from the earth are obtained from x, y, and after that we proceed agreeably
to some one of the five methods of articles 124-128. In fact, the symbols ?], if
being also taken with the meaning of article 128, and, analogously, the quotient
arising from the division of the sector contained between the two radii vectores
by the area of the triangle between the same being denoted by rf, we shall have,
Hl — L i. !^ — ^ !L
Sect. 1.] three complete observations. 181
and it readily appears, tliat if n, n', w", are regarded as small quantities of the first
order, i^ — 1> v' — ^> V" — ^ ^^^> generally speaking, quantities of the second
order, and, therefore,
6" 6"
the approximate values of x,t/, differ from the true ones only by quantities
of the second order. Nevertheless, upon a nearer examination of the sub-
ject, this method is found to be wholly unsuitable ; the reason of this we
will explain in a few words. It is readily perceived that the quantity (0. 1. 2),
by which the distances in the formulas 9, 10, 11, of article 114 have been multi-
plied, is at least of the third order, while, for example, in equation 9 the quan-
tities (0. 1. 2), (1. 1. 2), (II. 1. 2), are, on the contrary, of the first order; hence,
it readily follows, that an error of the second order in the values of the quanti-
ties -., "-r produces an error of the order zero in the values of the distances.
Wherefore, according to the common mode of speaking, the distances would be
affected by a finite error even when the intervals of the times were infinitely
small, and consequently it would not be admissible to consider either these dis-
tances or the remaining quantities to be derived from them even as approximate ;
and the method would be opposed to the second condition of the preceding
article.
132.
Putting, for^the sake of brevity,
(0.1.2) = «, (O.L2)ir = — J, (0.O.2)Z) = + e, (0.n.2)i)'' = + c?,
so that the equation 10, article 114, may become
US' ^=h -\-c —r A- d —r,
the coefficients c and d wHl, indeed, be of the first order, but it can be easily
shown that the difference c — d is to be referred to the second order. Then it
follows, that the value of the quantity
cn-\-dn''
n-\-n"
182 DETERMINATION OF AN ORBIT FROM [BoOK IL
resulting from tlie approximate assumption that n : i{' = ^ : ^" is affected by an
error of the fourth order only, and even of the fifth only when the middle is dis-
tant from the extreme observations by equal intervals. For this error is
where the denominator is of the second order, and one factor of the numerator
&^" {d — c) of the fourth, the other r(' — ri of the second, or, in that special case,
of the third order. The former equation, therefore, being exhibited in this form,
r,/ , I c n -\- d v!' n -\- n"
ad =^b-{-- \ „ .—h—,
' n-\-n n '
it is evident that the defect of the method explained in the preceding article does
not arise from the fact that the quantities w, i{' have been assumed proportional to
^, ^", but that, in addition to this, n' was put propoii;ional to ^'. For, indeed, in this
way, instead of the factor ^~^" , the less exact value "^ = 1 is introduced,
from which the true value
•^ 2 rjt/'rrV cosf cos f cos f"
differs by a quantity of the second order, (article 128).
133.
Since the cosines of the angles/,/',/", as also the quantities rj, if differ from
unity by a difference of the second order, it is evident, that if instead of
n'
the approximate value
14- "^
is introduced, an error of the fourth order is committed. If, accordingly, in place
of the equation, article 114, the following is introduced,
an error of the second order will show itself in the value of the distance d' when
Sect. 1.] three complete observations. 183
the extreme observations are equidistant from the middle ; or, of the first order in
other cases. But this new form of that equation is not suited to the determina-
tion of d', because it involves the quantities r, r', r", still unknown.
Now, generally speaking, the quantities ^ , ■;7 , differ from unity by a quantity
of the first order, and in the same manner also the product ^: it is readily
perceived that in the special case frequently mentioned, this product differs
from unity by a quantity of the second order only. And even when the orbit
of the ellipse is slightly eccentric, so that the eccentricity may be regarded as a
quantity of the first order, the difierence of ^ can be referred to an order one
degree higher. It is manifest, therefore, that this error remains of the same order
as before if, in our equation, ^^^^, is substituted for g^, whence is obtained the
following form,
In fact, this equation still contains the unknown quantity /, which, it is evident
nevertheless, can be eliminated, since it depends only on d' and known quantities.
K now the equation should be afterwards properly arranged, it would ascend to
the eighth degree.
134.
From the preceding it wiU be understood why, in our method, we are about
to take for x, y, respectively, the quantities
i"=P, and 2 (^"- !)/»=«.
For, in the first place, it is evident that if P and Q are regarded as known quanti-
ties, 8' can be determined from them by means of the equation
and afterwards d,d'', by equations 4, 6, article 114, since we have
In the second place, it is manifest that y , ^ ^" are, in the first hypothesis, the
184 DETERMINATION OF AN ORBIT FROM [BoOK 11.
obvious approximate values of the quantities P, Q, of which the true values are
precisely
6 if" rr"tj7j" cos f cos f cos f" '
from vi^hich hypothesis will result errors of the first order in the determination of
d', and therefore of d, d", or of the second order in the special case several times
mentioned. Although we may rely with safety upon these conclusions, generally
speaking, yet in a particular case they can lose their force, as when the quantity
(0. 1. 2), which in general is of the third order, happens to be equal to zero, or so
small that it must be referred to a higher order. This occurs when the geocentric
path in the celestial sphere has a point of contrary flexure near the middle place.
Lastly, it appears to be required, for the use of our method, that the heliocentric
motion between the three observations be not too great : but this restriction, by
the nature of the very complicated problem, cannot be avoided in any way;
neither is it to be regarded as a disadvantage, since it will always be desired to
begin at the earliest possible moment the first determination of the unknown
orbit of a new heavenly body. Besides, the restriction itself can be taken in a
sufficiently broad sense, as the example to be given below will show.
135.
The preceding discussions have been introduced, in order that the principles
on which our method rests, and its true force, as it were, may be more clearly
seen : the practical treatment, however, will present the method in an entirely
different form which, after very numerous applications, we can recommend as
the most convenient of many tried by us. Since in determining an unknown
orbit from three observations the whole subject may always be reduced to
certain hypotheses, or rather successive approximations, it will be regarded as a
great advantage to have succeeded in so arranging the calculation, as, at the
beginning, to separate from these hypotheses as many as possible of the compu-
tations which depend, not on P and Q, but only on a combination of the known
quantities. Then, evidently, these preliminary processes, common to each hypoth-
esis, can be gone through once for all, and the hypotheses themselves are reduced
Sect. 1.] three complete observations. 185
to the fewest possible details. It will be of equally great importance, if it
should not be necessary to proceed in every hypothesis as far as the elements,
but if their computation might be reserved for the last hypothesis. In both
these respects, our method, which we are now about to explain, seems to leave
nothing to be desired.
136.
We are, in the first place, to connect by great circles three heliocentric places
of the earth in the celestial sphere. A, A', A" (figure 4), with three geocentric
places of the heavenly body, B, B', B", and then to compute the positions of these
great circles with respect to the ecliptic (if we adopt the ecliptic as the funda-
mental plane), and the places of the points B, B' , B", in these circles.
Let a, a\ a" be three geocentric longitudes of the heavenly body, /5, ^', /5", lat-
itudes ; /, t, I", heliocentric longitudes of the earth, the latitudes of which we put
equal to zero, (articles 117, 72). Let, moreover, /, y', y" be the incHnations to the
echptic of the great circles drawn from A, A!., A!.', to B, B', B", respectively ; and,
in order to follow a fixed rule in the determination of these inclinations, we shall
always measure them from that part of the ecliptic which lies in the direction
of the order of the signs from the points A, A', A", so that their magnitudes will
be counted from 0 to 360°, or, which amounts to the same thing, from 0 to 180°
north, and from 0 to — 180° south. We denote the arcs AB, AB', A'B", which
may always be taken between 0 and 180°, by d,d', d". Thus we have for the de-
termination of y and d the formulas,
n -I J tan
[1] tany=-.
sm(« — I)
[2] i^nd=.'^^^^^^^^.
To which, if desirable for confirming the calculation, can be added the following,
sin(^ = ^^, cos(5' = cos/3 cos(a — T).
sin 7' ' ^ ''
We have, evidently, entirely analogous formulas for determining y', d\ y", d". Now,
if at the same time /5 = 0, cf — ^= 0 or 180°, that is, if the heavenly body should
24
186 DETERMINATION OF AN ORBIT FROM [BoOK II.
be in opposition or conjunction and in the ecliptic at the same time, y would be
indeterminate. But we assume that this is not the case in either of the three
observations.
If the equator is adopted as the fundamental plane, instead of the ecliptic,
then, for determining the positions of the three great circles with respect to the
equator, will be required the right ascensions of their intersections with the equa-
tor, besides the inclinations ; and it will be necessary to compute, in addition to
the distances of the points B, B', B", from these intersections, the distances of the
points A, A, A" also from the same intersections. Since these depend on the
problem discussed in article 110, we do not stop here to obtain the formulas.
137.
The second step will be the determination of the positions of these three great
circles relatively to each other, which depend on their inclinations and the places
of their mutual intersections. If we wish to bring these to depend upon clear
and general conceptions, without ambiguity, so as not to be obliged to use
special figures for different individual cases, it will be necessary to premise some
preliminary explanations. Firstly, in every great circle two opposite directions
are to be distinguished in some way, which will be done if we regard one of them
as direct or positive, and the other as retrograde or negative. This being wholly
arbitrary in itself, we shall always, for the sake of establishing a uniform rule, con-
sider the dhections from A, A', A" towards B, B', B" as positive \ thus, for example,
if the intersection of the first circle with the second is represented by a positive
distance from the point A, it will be understood that it is to be taken from A
towards B (as D" in our figure) ; but if it should be negative, then the distance
is to be taken on the other side of A. And secondly, the two hemispheres, into
which every great circle divides the whole sphere, are to be distinguished by suit-
able denominations; accordingly, we shall call that the S2<^e;w hemisphere, which,
to one walking on the inner surface of the sphere, in the positive direction along
the great circle, is on the right hand ; the other, the inferior. The superior hemi-
sphere will be analogous to the northern hemisphere in regard to the ecliptic or
equator, the inferior to the southern.
Sect. 1.] three complete observations. 187
These definitions being correctly understood, it will be possible conveniently
to distinguish both intersections of the two great circles from each other. In fact,
in one the first circle- tends from the inferior to the superior hemisphere of the
second, or, which is the same thing, the second from the superior to the inferior
hemisphere of the first ; in the other intersection the opposite takes place.
It is, indeed, wholly arbitrary in itself which intersections we shall select for
our problem ; but, that we may proceed here also according to an invariable rule,
we shall always adopt these (Z>, D\D", figure 4) where the third circle A'B" passes
into the superior hemisphere of the second AB\ the third into that of the first
AB, and the second into that of the first, respectively. The places of these inter-
sections will be determined by their distances from the points A and A', A and
A', A and A, which we shall simply denote by AD, A'D, AD', A'D\ AD", AD".
Which being premised, the mutual inclinations of the circles will be the angles
which are contained, at the points of intersection D, D', D", between those parts
of the circles cutting each other that lie in the positive direction ; we shall
denote these inclinations, taken always between 0 and 180°, by e, &', h". The de-
termination of these nine unknown quantities from those that are known, evi-
dently rests upon the problem discussed by us in article 55. We have, conse-
quently, the following equations : —
[3] sin ^ £ sin ^ {AD -\- A'D) = sin i {I" — T) sin i {y" + y'\
[4] sin J £ cos ^ {AD + A'D) = cos i {f — I') sin h {f — 7'),
[5] cos I £ sin I {AD — A'D) = sin h {l" — I') cos h {/' + /),
[6] cos i £ cos ^ {A'D — A'D) = cos ^ {f — I') cos i {/' — /).
i {AD-\-A"D) and sin ^ £ are made known by equations 3 and 4, i {AD — A'D)
and cos I- e by the remaining two ; hence A'D, A'D and £. The ambiguity in the
determination of the arcs i {A'D -\- A'D), i {A'D — A'D), by means of the tan-
gents, is removed by the condition that sin i e, cos ^ £, must be positive, and the
agreement between sin ^ £, cos -k e, will serve to verify the whole calculation.
The determination of the quantities AD', A'D', a', AD", AID', t" is efiected in
precisely the same manner, and it will not be worth while to transcribe here the
eight equations used in this calculation, since, in fact, they readily appear if we
change
188
DETERMINATION OF AN ORBIT FROM
[Book H.
AD
A'D
£
r—t
y"
/
forJLTr
A'D'
e'
i"—i
f
Y
or for AD"
AD"
f."
I' —I
7'
7
respectively.
A new verification of the whole calculation thus far can be obtained from the
mutual relation between the sides and angles of the spherical triangle formed by
joining the three points D, D', D", from which result the equations, true in gen-
eral, whatever may be the positions of these points,
%m{A]y — AD') _ sm {A'D — A'D') _ sin (A"D — A"D')
sin £ sin a' sin n"
Finally, if the equator is selected for the fundamental plane instead of the eclip-
tic, the computation undergoes no change, except that it is necessary to sub-
stitute for the heliocentric places of the earth A, A, A" those points of the equa-
tor where it is cut by the circles AB, AB', A'B" ; consequently, the right ascen-
sions of these intersections are to be taken instead of I, X, V, and also instead of
AD, the distance of the point D from the second intersection, etc.
138.
The tliird %iQ.^ consists in this, that the two extreme geocentric places of the
heavenly body, that is, the points B, B", are to be joined by a great circle, and
the intersection of this with the great circle AB' is to be determined. Let B^ be
this intersection, and d' — a its distance from the point A ; let a* be its longitude,
and /:J=== its latitude. We have, consequently, for the reason that B, B^-, B" lie in
the same great circle, the well-known equation,
0 = tan /5 sin {a" — a=-=) — tan /5'== sin {a" — «) + tan /5" sin (a* — «),
which, by the substitution of tan y' sin (a* — /') for tan /5% takes the following
form : —
0 = cos (a* — t) (tan /? sin {a" — t) — tan ^" sin (« — f ))
— sin («=== — /') (tan /5 cos [a" — 1)-\- tan / sin {a" — a) — tan /5" cos (a — T)).
Wherefore, since tan («■•= — t) = cos y' tan {d' — o) we shall have,
tan(d' — ff) =
tan ^ sin (a" — l') — tan ^' sin (cc — I')
cos / (tan § cos (a" — V) — tan ^' cos (a — 0) + ^i° 7' ^^^ ("" — ^) '
Sect. 1.] three complete observations. 189
Thence are derived the following formulas, better suited to numerical calculations.
Putting,
[7] tan (i sin (a" — l') — tan /5'' sin {a — /') = JSf,
[8] tan ^ cos {a'' — V) — tan (S" cos (« — I') = Tsuit,
[9] sm{a" — a) = Tcost,
we shall have (article 14, IT.)
[10] UniS'-a) = ^^^y
The uncertainty in the determination of the arc {d' — a) by means of the
tangent arises from the fact that the great circles A'B', BB", cut each other in
tivo points ) we shall always adopt for B^ the intersection nearest the point B' , so
that o may always fall between the limits of — 90° and -\- 90°, by which means
the uncertainty is removed.
For the most part, then, the value of the arc a (which depends upon the
curvature of the geocentric motion) will be quite a small quantity, and even, gen-
erally speaking, of the second order, if the intervals of the times are regarded
as of the first order.
It wiU readily appear, from the remark in the preceding article, what are the
modifications to be applied to the computation, if the equator should be chosen
as the fundamental plane instead of the ecliptic. It is, moreover, manifest that
the place of the point B^- will remain indeterminate, if the circles ^i>",J.'^"
should be wholly coincident ; this case, in which the four points J.', B, B\ B" lie in
the same great circle, we exclude from our investigation. It is proper in the
selection of observations to avoid that case, also, where the locus of these four
points differs but little from a great circle ; for then the place of the point B^,
which is of great importance in the subsequent operations, would be too much
affected by the slightest errors of observation, and could not be determined with
the requisite precision. In the same manner the point B^, evidently, remains
indeterminate when the points B, B" coincide,-]- in which case the position of the
t Or when they are opposite to each other ; but we do not speak of this case, because our method is
not extended to observations embracing so great an interval.
190 DETERMINATION OF AN ORBIT FROM [BoOK II.
circle BB" itself would become indeterminate. Wherefore we exclude this case,
also; just as, for reasons sunilar to the preceding, those observations will be
avoided in which the first and last geocentric places fall in points of the sphere
near to each other.
139.
Let C, C, C", be three heliocentric places of the heavenly body in the celestial
sphere, which will be (article 64, III.) in the great circles AB, AB', A'B", respec-
tively, and, indeed, between A and B, A and B', A' and B" ; moreover, the points
C, C, G" will lie in the same great chcle, that is, in the circle which the plane
of the orbit projects on the celestial sphere.
We will denote by r, r', r" , three distances of the heavenly body from the sun ;
by 9, ^, q", its distances from the earth ; by R, R, R", the distances of the earth
from the sun. Moreover, we put the arcs CO", CO", CO' equal to 2/, 2f, 2f",
respectively, and
rV sin 2/ = n, rr" sin 2/ = ?/, r r sin 2/' = ii'.
Consequently we have
f=f-\.f", AO-]-CB = d, AO'-[-0'B' = d', A'0"-^0"B" = d";
also,
sinS sin^C sin (75
r Q R
sin 8' sin A' G' sin O'B'
~?~~ 7 ~ R'
sin S' _ sin J!' G" _ sin G"B"
/' ~~ (J' ~ S' '
Hence it is evident, that, as soon as the positions of the points 0, C, G" are known,
the quantities r, /, r", q, ()', q" can be determined. We shall now show how the
former may be derived from the quantities
i=P,2(»-+=:-l)r"=«,
from which, as we have before said, ovu: method started.
Sect. 1.] three complete observations. 191
140.
We first remark, that if N were any point whatever of the great circle CC C"^
and the distances of the points C, C, G" from the point N were counted in the
direction from C to C", so that in general
NC" — NC'=2f, NG" — NG=^2f\ NG' — ]SfG=2f",
we shall have
I. 0 = sin 2/sin NG— sin 2/ sin NG' + sin 2f" sin NG".
We will now suppose N to be taken in the intersection of the great circles
BB^B", CC" C", as in the ascending node of the former on the latter. Let us
denote by (S, (£', S", 2), 2)', 2)", respectively, the distances of the points G, G', G",
D, D', D" from the great circle BB^B", taken positively on one side, and nega-
tively on the other. Then sin d, sin %!, sin S", will evidently be proportional to
siniVC, siniVC, siniVC"', whence equation I. is expressed in the following form: —
0 = sin 2/ sin S — sin 2/ sin £' + sin If" sin ^' ;
or multiplying by rr/',
n. 0 — nr sin ^ — nr sin g' -f ri'r" sin S".
It is evident, moreover, that sin £ is to sin '3)', as the sine of the distance of the
point G from B is to that of D' from B, both distances being measured in the
same direction. We have, therefore,
. a' sin %' sin GS
sm (^AD — Gy
in precisely the same way, are obtained,
. ^ sin ID" sin OS
— sin ^ :
— sinS":
sin X sin C'B* sm.lj' sm G' B*
'' (sTn^'Z)— 3' + (t) sin(^'i7'— S' + ff)'
sin X sin G"B" sin %' sin G"B'
sin {A'D — 5") ~ sin {A'U — 6") *
Dividing, therefore, equation IT. by r" sin (E", there results,
~ ^^ ' /'sin G"B" ' sin (AD' —d) ^ ' 7^'smG"B" ' sin (A'B—S'-^-a)
192 DETERMINATION OF AN ORBIT FROM [BoOK II.
If now we designate the arc G'B' by z, substitute for r, /, /' their values in
the preceding article, and, for the sake of brevity, put
[11]
JR' sm.^' €ra.{Aiy ■
L-^^-J R' sin 5" sin {AD _ 3' + a) ~~ *'
our equation wiU become
m. 0 = an — ^^/?^^4:=l^+n"
smz '
The coefl&cient I may be computed by the following formula, which is easil;^
derived from the equations just introduced : —
ig^sina'sin(^i7^ — 5) _ ,
L15J a X ^ gjjj 3 gij^ {A!D'—b' + (t) — ^ •
For verifying the computation, it wiU be expedient to use both the formulas 12
and 13. When sin(J['Z>" — ^' -[-<?) is greater than sin(J.'i> — d' -^-q), the latter
formula is less affected by the unavoidable errors of the tables than the former,
and so wiU be preferred to it, if some small discrepancy to be explained in this
way should result in the values of i ; on the other hand, the former formula is
most to be relied upon, when sin [A^D" — d' -\- o) is less than sin [AD — d' -\- o);
a suitable mean between both values will be adopted, if preferred. The follow-
ing formulas can be made to answer for examining the calculation; their not very
difl&cult derivation we suppress for the sake of brevity.
p. asin(Z" — V) hsm{l" — I) sin (5' — a) |^ sin (Z' — I)
B R' • si^'fi^ I ~E' '
, R' sin 5' fTcos R cos S'
~~ R" sin 8"* sin {A U — 8) sin «"
in which (article 138, equation 10,) U expresses the quotient
S _r sin (<-!-/)
sin ((5' — a) cos {d' — a) '
141.
From P = — , and equation HI. of the preceding article, we have
, , „.P-\-a I ,sin(2: — a)
Sect. 1.] theee complete observations. 193
thence, and from
_ 2 (^' - ly and /= ^^
\ «' / sinz
is obtained,
Putting, therefore, for the sake of brevity,
L-^ -1 2E'^sm^8'smG~^'
and introducing the auxiliary angle w such that
tan CD =
3^^ — cosa,
we have the equation
IV. e Qsiao) sin* s = sin (^ — w — or),
from which we must get the unknown quantity s. That the angle w may be
computed more conveniently, it will be expedient to present the preceding for
mula for tan w thus : —
P(- ^) + (- «)
Vcosff / ' Vcosff /
"Whence, putting.
d,
we shall have for the determination of to the very simple formula,
tana>-^^^+^
We consider as the fourth step the computation of the quantities a, b, c, d, e,
25
[16]
[16]
a
cos a
' 1
tanff
* 1
194 DETERiirXATION OF AN OEBIT FROM [BoOK 11.
by means of the formulas 11-16, depending on given quantities alone. The
quantities h, c, e, will not themselves be required, only their logarithms.
There is a special case in which these precepts require some change. That
is, when the great circle BB" coincides with A'B", and thus the points B, B'^
with D', D, respectively, the quantities a, h would acquire infinite values. Put-
ting, in this case,
^ sin d sin {A'jy — S'J^a)
R'smd's\n{A]y' — d) ~^'
in place of equation HE. we shall have
f. n' sin (z — d)
sm z '
whence, making
, jr sin ff
tan 0) = p I ., r,
P-{-(l — ncosa)^
the same equation IV. is obtained.
In the same manner, in the special case when g = 0, c becomes infinite, and
w = 0, on account of which the factor c sin w, in equation IV., seems to be inde-
terminate ; nevertheless, it is in reality determinate, and its value is
J P + g
2R'HmH' {b—l){P^d)'
as a little attention wiU show. In this case, therefore, sm s becomes
142.
Equation IV., which being developed rises to the eighth degree, is solved by
trial very expeditiously in its unchanged form. But, from the theory of equa-
tions, it can be easily shown, (which, for the sake of brevity, we shall dispense
with explaining more fully) that this equation admits of two or four solutions by
means of real values. In the former case, one value of sin s will be positive ;
and the other negative value must be rejected, because, by the nature of the
pro])lem, it is impossible for / to become negative. In the latter case, among the
values of sin s one will be positive, and the remaining three negative, — when.
Sect. 1.] three complete observations. 195
accordingly, it will not be doubtful which must be adopted, — or three positive
with one negative ; in this case, from among the positive values those, if there are
any, are to be rejected which give s greater than d', since, by another essential
condition of the problem, {/ and, therefore, sin {d' — g), must be a positive quantity.
When the observations are distant from each other by moderate intervals of
time, the last case will most frequently occur, in which three positive values of
sin 0 satisfy the equation. Among these solutions, besides that which is true,
some one will be found making 0 differ but little from d\ either in excess or
in defect; this is to be accounted for as follows. The analytical treatment of
our problem is based upon the condition, simply, that the three places of the heav-
enly body in space must fall in right lines, the positions of which are determined
by the absolute places of the earth, and the observed places of the body. Now,
from the very nature of the case, these places must, in fact, fall in those parts of
the right lines whence the light descends to the earth. But the anatytical equa-
tions do not recognize this restriction, and every system of places, harmonizing of
course with the laws of Kepler, is embraced, whether they lie in these right lines
on this side of the earth, or on that, or, in fine, whether they coincide with the
earth itself Now, this last case will undoubtedly satisfy our problem, since the
earth moves in accordance with these laws. Thence it is manifest, that the equa-
tions must include the solution in which the points C. C, G" coincide with A, A', A"
(so long as we neglect the very small variations in the elliptical places of the earth
produced by the perturbations and the parallax). Equation TV., therefore, must
always admit the solution s = d', if true values answering to the places of the
earth are adopted for P and Q. So long as values not differing much from these
are assigned to those quantities (which is always an admissible supposition, when
the intervals of the times are moderate), among the solutions of equation TV.,
some one will necessarily be found which approaches very nearly to the value
For the most part, indeed, in that case where equation IT. admits of three
solutions by means of positive values of sin s, the third of these (besides the true
one, and that of which we have just spoken) makes the value of 2 greater than
d', and thus is only analytically possible, but physically impossible ; so that it can-
196 DETERMINATION OF AN ORBIT FROM [BoOK II.
not then be doubtful wbicb is to be adopted. But yet it certainly can happen,
that the equation may admit of two distinct and proper solutions, and thus that
our problem may be satisfied by two wholly different orbits. But in such an
event, the true orbit is easily distinguished from the false as soon as it is possible
to bring to the test other and more remote observations.
143.
As soon as the angle z is got, r is immediately had by means of the equation
r = ; .
sinz
Further, from the equations P = — and III. we obtain,
n'r' {P-\-d)R'%m8'
n b sin (z — a) '
nV 1 nV
n" P ' n
Now, in order that we may treat the formulas, according to which the posi-
tions of the points C, C", are determined from the position of the point C , in such
a manner that their general truth in those cases not shown in figure 4 may
immediately be apparent, we remark, that the sine of the distance of the point
C from the great circle CB (taken positively in the superior hemisphere, nega-
tively in the inferior) is equal to the product of sin ^' into the sine of the distance
of the point C from Z>", measured in the positive direction, and therefore to
— sin ^" sin CD" = — sin e" sin (s -j- A'D" — d') ;
in the same manner, the sine of the distance of the point 0" from the same great
circle is — sin e' sin C"JD\ But, evidently, those sines are as sin CO' to sin CO", or
as ^, to ^, , or as ti'V to nV. Putting, therefore, CD' = ^", we have
r smi, =z—^.- — ^sm(s-\- AD — o).
n" sine ^ ' ^
Precisely in the same way, putting CD' = t, is obtained
TT-T • ;- ^''^ sine • / i a't, s>/\
VI. ;• sni (, = — . ^-7 sni (s -4- AD — o ) .
n sme ^ ' '
Vn. ■ r sin (C + AD" — AD') = r" P |^ sin (^ -f Al'D — A'D').
Sect. 1.] three complete observations. 197
By combining equations Y. and VI. with the following taken from article 139,
Vin. r" sin (C'' — ^'D' + §") = R" sin d",
IX. r&m{(; — AD'^§) = R^md,
the quantities t, ^", r, r'jwill be thence derived by the method of article 78.
That this calculation may be more conveniently effected, it wiU not be unaccept-
able to produce here the formulas themselves. Let us put
i?sin d
[17]
[18]
sin {AD' — (
E'^md"
&m{^A'D' — d")
The computation of these, or rather of their logarithms, yet independent of P
and Q, is to be regarded as the ffth and last step in the, as it were, preliminary
operations, and is conveniently performed at the same time with the computation
of a, b, themselves, or with the fourth step, where a becomes equal to ~ .
Making, then,
n sine
n'r' sin f'
«" * sin £
^, sin [z + A'D — d') =p,
^{lp—l) = q,
we derive C and r from r sin ^ = j??, r cos ^ = ^ ; also, ^" and r" from r" sin <i" =^p",
and r" cos i^" = q". No ambiguity can occur in determining t and 'C'\ because r
and /' must, necessarily, be positive quantities. The complete computation can,
if desired, be verified by equation YII.
There are two cases, nevertheless, where another course must be pursued.
That is, when the point D' coincides with B, or is opposite to it in the sphere,
or when AD' — d = 0 or 180°, equations VI. and IX. must necessarily be iden-
198 DETERailNATION OF AN ORBIT FROM [BoOK 11.
tical, and we should have k = oo , Ip — 1 = 0, and q, therefore, indeterminate.
In this case, I," and ;•" will be determined, in the manner we have shown, but
then C and r must be obtained b j the combination of equation YII. with VI. or
IX. We dispense with transcribing here the formulas themselves, to be found
in article 78; we observe, merely, that in the case where AD' — d" is in fact
neither =: 0 nor = 180°, but is, nevertheless, a very small arc, it is preferable
to follow the same method, since the former method does not then admit of the
requisite precision. And, in fact, the combination of equation VII. with VI. or IX.
will be chosen according as sin [AD" — AD') is greater or less than sin [AD' — d).
In the same manner, in the case in which the point D', or the one opposite to
it, either coincides with B" or is little removed from it, the determination of C"
and 7'' by the preceding method would be either impossible or unsafe. In this
case, accordingly, t and r will be determined by that method, but l" and r" by
the combination of equation VII. either with V. or with VIII., according as sin
[A'D — A'D') is greater or less than sin {A'D^ — d").
There is no reason to fear that D' wiU coincide at the same time with the points
B, B", or with the opposite points, or be very near them ; for the case in which
B coincides with B", or is but little remote from it, we excluded above, in article
138, from our discussion.
144.
The arcs t, and L." being found, the positions of the points C, C", will be given,
and it will be possible to determine the distance CC"-= 2/' from C, l" and t.
Let u, u", be the inclinations of the great circles AB, A"B" to the great circle CO"
(which in figure 4 will be the angles COD' and 180° — C6'"Z>', respectively),
and we shall have the following equations, entirely analogous to the equations
3-6, article 137 : —
sin/' sin ^ (?f" -j- tt) ^ sin I e' sin ^ (t -|- ^"),
sin/' cos ^ {u" -j- m) = cos ^ e' sin ^c — (^"),
cos/' sin i (u" — u) =z sin ^ e' cos 2 (C -[- C")>
cos/' cos i (u" — u) = cos I e' cos i (; — C").
Sect. 1.] three complete observations. 199
The two former will give ^ {iif'-\-i(,) and sin/', the two latter ^ (^«" — u) and cos/';
from sin/' and cos/' we shall have/'. It will be proper to neglect in the first
hypotheses the angles ^ (?i!"-|-^0 and i [u" — u), which will be used in the last
hypothesis only for determining the position of the plane of the orbit.
In the same way, exactly,/ can be derived from £, CD and C"D; also/"
from e", CD" and O'D" ; but the following formulas are used much more con-
veniently for this purpose : —
sin 2 / = r sin 2 /' . -7-7,
sin 2 /" = r" sin 2/'.'^,
. . n ri'
in which the logarithms of the quantities -7-;, -7—, are already given by the pre-
ceding calculations. Finally, the whole calculation finds a new verification in
this, that we must have
2/-|-2/" = 2/';
if by chance any difference shows itself, it will not certainly be of any impor-
tance, if all the processes have been performed as accurately as possible. Never-
theless, occasionally, the calculation being conducted throughout with seven
places of decimals, it may amount to some tenths of a second, which, if it appear
worth while, we may with the utmost facility so distribute between 2 /and 2/"
that the logarithms of the sines may be equally either increased or diminished,
by which means the equation
p r sin 2/' w"
/'sin2/~^
will be satisfied with all the precision that the tables admit. When /and/" differ a
little, it will be sufficient to distribute that difference equally between 2/ and 2/".
145.
After the positions of the heavenly body in the orbit have been determined in
this manner, the double calculation of the elements will be commenced, both by
the combination of the second place with the third, and the combination of the
first with the second, together with the corresponding intervals of the times.
200 DETERMINATION OF AN ORBIT FROM [BoOK II.
Before this is undertaken, of course, the intervals of the times themselves require
some correction, if it is decided to take account of the aberration agreeably to the
third method of article 118. In this case, evidently, for the true times are to be
substituted fictitious ones anterior to the former, respectively, by 493(^), 493^',
493^)" seconds. For computing the distances q, q', i>", we have the formulas : —
_ Esm(AI>' — t) _rsm(Aiy — 0
^'~sm{^—AD'-\-6)~ smd '
, Ii'sm(d'—s) r' sin (S'—z)
" sin z sin ^ '
,,_ m' sin (A"iy — C) _ /' sin (A" IT — C)
^ ~sm{Z" — A"iy-\-d")~ smd"
But, if the observations should at the beginning have been freed from
aberration by the first or second method of article 118, this calculation may be
omitted ; so that it will not be necessary to deduce the values of the distances (>,
(^/, q", unless, perhaps, for the sake of proving that those values, upon which the
computation of the aberration was based, were sufficiently exact. Finally, it is
apparent that all this calculation is also to be omitted whenever it is thought
preferable to neglect the aberration altogether.
146.
The calculation of the elements — on the one hand from /, r", 2/ and the
corrected interval of the time between the second and third observations, the
product of which multiplied by the quantity k, (article 1,) we denote by &, and
on the other hand from r, /, If" and the interval of time between the first and
second observations, the product of which by Ti will be equal to ^" — is to be car-
ried, agreeably to the method explained in articles 88-105, only as far as the
quantity there denoted by y, the value of which in the first of these combinations
we shall call iq, in the latter r['. Let then
Orf ' r 7^' rirf' cos f cos f cos f" '
and it is evident, that if the values of the quantities P, Q, upon which the whole
calculation hitherto is based, were true, we should have in the result P' = P,
Sect. 1.] three complete observations. 201
Q' ■= Q. And conversely it is readily perceived, that if in the result P' = P,
Q' = Q, the double calculation of the elements from both combinations Avould, if
completed, furnish numbers entirely equal, by which, therefore, all three observa-
tions will be exactly represented, and thus the problem wholly satisfied. But
when the result is not P' = P, Q'= Q, let P'—P, Q' — Q be taken for X and Y,
if, indeed, P and Q were taken for x and y; it will be still more convenient to put
logP = ^,log ^ = ^,logP'— logP = X,log ^'_log ^= Z ,
Then the calculation must be repeated with other values of x, y.
147.
Properly, indeed, here also, as in the ten methods before given, it would be
arbitrary what new values we assume for x and y in the second hypothesis, if
only they are not inconsistent with the general conditions developed above ; but
yet, since it manifestly is to be considered a great advantage to be able to set out
from more accurate values, in this method we should act with but little prudence
if we were to adopt the second values rashl}^, as it were, since it may easily be
perceived, from the very nature of the subject, that if the first values of P and Q
were affected with slight errors, P' and Q' themselves would represent much more
exact values, supposing the heliocentric motion to be moderate. Wherefore, we
shall always adopt P' and Q' themselves for the second values of P and Q, or
log P', log Q' for the second values of x and y, if log P, log Q are supposed to
denote the first values.
Now, in this second hypothesis, where all the preliminary work exhibited
in the formulas 1-20 is to be retained without alteration, the calculation will be
undertaken anew in precisely the same manner. That is, first, the angle o>
will be determined; after that z, r, ^, !!^, c, r, i;", r", f, f, f". From the dif-
ference, more or less considerable, between the new values of these quantities
and the first, a judgment will easily be formed whether or not it is worth while
to compute anew the correction of the times on account of aberration ; in the
latter case, the intervals of the times, and therefore the quantities & and &", wiU
remain the same as before. Finally, i], if are derived from /, r, r"J", r, r and
26
202 DETERMINATION OF AN ORBIT FROM [BoOK 11.
the intervals of the times ; and hence new values of P' and Q', which commonly
differ much less from those furnished by the first hypothesis, than the latter from
the original values themselves of P and Q. The second values of X and Y will,
therefore, be much smaller than the first, and the second values of P', Q', will be
adopted as the third values of P, Q, and with these the computation will be
resumed anew. In this manner, then, as from the second hypothesis more exact
numbers had resulted than from the first, so from the third more exact numbers
will again result than from the second, and the third values of P', Q' can be taken
as the fourth of P, Q, and thus the calculation be repeated until an hypothesis
is arrived at in which X. and Y may be regarded as vanishing ; but when the
third hypothesis appears to be insufficient, it will be preferable to deduce the val-
ues of P, Q, assumed in the fourth hypothesis from the first three, in accordance
with the method explained in articles 120, 121, by which means a more rapid
approximation will be obtained, and it will rarely be requisite to go forward to
the fifth hypothesis.
148.
When the elements to be derived from the three observations are as yet
wholly unknown (to which case our method is especially adapted), in the first
hypothesis, as we have already observed, — , ^ d", are to be taken for approximate
values of P and Q, where ^ and ^" are derived for the present from the intervals
of the times not corrected. If the ratio of these to the corrected intervals is
expressed by /^ : 1 and u" : 1, respectively, we shall have in the first hypothesis,
X = log fii — log //' -|- log tj — log rj'^
Y= log u -\- log jLi" — log 1] — log if -\- Comp. log cos/ -\- Comp. log cos /'
-J- Comp. log cos/" -|- 2 log r — log r — log r".
The logarithms of the quantities ,a, fjf', are of no importance in respect to the re-
maining terms ; log 7^ and log ?j", which are both positive, in X cancel each other
in some measure, whence X possesses a small value, sometimes positive, some-
times negative ; on the other hand, in Y some compensation of the positive terms
Comp. log cos/, Comp. log cos/', Comp. log cos/" arises also from the negative
Sect. 1.] three complete observations. 203
terms log i], log rf', but less complete, for the former greatly exceed the latter. In
general, it is not possible to determine any thing concerning the sign of log -^.
Now, as often as the heliocentric motion between the observations is small, it
will rarely be necessary to proceed to the fourth hypothesis ; most frequently the
third, often the second, will afford sufficient precision, and we may sometimes be
satisfied with the numbers resulting from even the first hypothesis. It will be
advantageous always to have a regard to the greater or less degree of precision
belonging to the observations; it would be an ungrateful task to aim at a pre-
cision in the calculation a hundred or a thousand times greater than that which
the observations themselves allow. In these matters, however, the judgment is
sharpened more by frequent practical exercise than by rules, and the skilful
readily acquire a certain faculty of deciding where it is expedient to stop.
149.
Lastly, the elements themselves will be computed in the final hypothesis,
either from/, r', r", or from/", r, /, carrying one or the other of the calculations
through to the end, which in the previous hypotheses it had only been requisite
to continue as far as rj^rj"; if it should be thought proper to finish both, the
agreement of the resulting numbers will furnish a new verification of the whole
work. It is best, nevertheless, as soon as /,/',/", are got, to obtain the elements
from the single combination of the first place with the third, that is, from f^,r, r",
and the interval of the time, and finally, for the better confirmation of the com-
putation, to determine the middle place in the orbit by means of the elements
found.
In this way, therefore, the dimensions of the conic section are made known,
that isy the eccentricity, the semi-axis major or the semi-parameter, the place
of the perihelion with respect to the heliocentric places C, C\ G" , the mean
motion, and the mean anomaly for the arbitrary epoch if the orbit is elliptical, or
the time of perihelion passage if the orbit is hyperbolic or parabolic. It only
remains, therefore, to determine the positions of the heliocentric places in the
orbit with respect to the ascending node, the position of this node with reference
to the equinoctial point, and the inclination of the orbit to the ecliptic (or the
204 DETERMINATION OF AN ORBIT FROil [BoOK II.
equator). AH this may be efifected by the solution of a single spherical tri-
angle. Let Q be the longitude of the ascending node ; i the inclination of the
orbit ; (/ and ff" the arguments of the latitude in the first and third observations ;
lastly, let I — 9,= h, I" — S2 = W. Calling, in figure 4, 9, the ascending node,
the sides of the triangle S^ilCwill be AD' — t, ^, li, and the angles opposite to
them, respectively, i, 180° — y,ii. We shall have, then,
sin i 2 sin i (^ -|- li) = sin ^ {AJ}' — Q sin 2 (/ -|- u)
sin h i cos 2 [g -\- h) = cos ^ [AD' — 'Q) sin ^ (7 — u)
cos h i sin ^ {g — h) = sin i (AI)' — C) cos 2 (/ -j- «)
cos J ^ cos ^ {g — h) = cos ^ [AD' — ;) cos i (/ — ^«).
The two first equations will give ^ (^-f-A) and sin ^ i, the remaining two ^ {g — h)
and cos ^ 2 ; from y will be known the jjlace of the perihelion with regard to the
ascending node, from h the place of the node in the ecliptic ; finally, ^ will be-
come known, the sine and the cosine mutually verifying each other. We can
arrive at the same object by the help of the triangle 9,A"C', in which it is only
necessary to change in the preceding formulas the symbols g, h, A, L, /, tt into g",
h", A", 'C", y", li' . That still another verification may be provided for the whole
work, it mil not be unserviceable to perform the calculation in both ways;
when, if any very slight discrepancies should show themselves between the values
of 2^ 9, and the longitude of the perihelion in the orbit, it will be proper to take
mean values. These difierences rarely amount to OM or 0^2, provided all the
computations have been carefully made with seven places of decimals.
When the equator is taken as the fmidamental plane instead of the ecliptic,
it will make no difference in the computation, except that in place of the points
A, A" the intersections of the equator with the great cu'cles AB, A.'B" are to be
adopted.
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
205
150.
We proceed now to the illustration of this method by some examples fully
explained, which will show, in the plainest manner, how generally it applies, and
how conveniently and expeditiously it leads to the desired result.*
The new planet Juno will furnish us the first example, for which purpose we
select the following observations made at Greenwich and communicated to us by
the distinguished Maskelyne.
Mean Time, Greenwich.
App. Eight Ascension.
App. Declination S.
1804, Oct. 5 10^ 51™ 6'
17 9 58 10
27 9 16 41
357° 10' 22".35
355 43 45 .30
355 11 10 .95
6° 40' 8"
8 47 25
10 2 28
From the solar tables for the same times is found
Longitude of the Sun
from App. Equin.
Nutation.
Distance from
the Earth.
Latitude of
the Sun.
Appar. Obhquity of
the Ecliptic.
Oct. 5
17
27
192° 28' 53".72
204 20 21 .54
214 16 52 .21
4- 15".43
4-15 .51
-|-15 .60
0.9988839
0.9953968
0.9928340 ■
— 0".49
+ 0.79
— 0.15
23° 27' 59".48
59.26
59 .06
We will conduct the calculation as if the orbit were wholly unknown : for
which reason, it will not be permitted to free the places of Juno from parallax,
but it will be necessary to transfer the latter to the places of the earth. Accord-
ingly we first reduce the observed places from the equator to the ecliptic, the
apparent obliquity being employed, whence results,
* It is incorrect to call one method more or less exact than another. That method alone can be con-
sidered to have solved the problem, by which any degree of precision whatever is, at least, attainable.
"Wherefore, one method excels another in this respect only, that the same degree of precision may be
reached by one more quickly, and with less labor, than by the other.
206
DETERMINATION OF AN ORBIT FROM
[Book II.
App. Longitude of Juno.
App. Latitude of Juno.
Oct. 5
17
27
354° 44' 54".27
352 34 44.51
351 34 51 .57
— 4°59'31".59
— 6 21 56.25
— 7 17 52.70
We join directly to this calculation the determination of the longitude and
latitude of the zenith of the place of observation in the three observations : the
right ascension, in fact, agrees with the right ascension of Juno (because the
observations have been made in the meridian) but the declination is equal to the
altitude of the pole, 51° 28' 39". Thus we get
Long, of the Zenith.
Lat. of the Zenith.
Oct. 5
17
27
24° 29'
23 25
23 1
46° 53'
47 24
47 36
Now the fictitious places of the earth in the plane of the ecliptic, from which
the heavenly body would appear in the same manner as from the true places of
the observations, will be determined according to the precepts given in article 72.
In this way, putting the mean parallax of the sun equal to 8".6, there results,
Reduction of Longitude.
Beduction of Distance.
Reduction of Time.
Oct. 5
17
27
— 22" .39
— 27 .21
— 35 .82
-1-0.0003856
4- 0.0002329
4- 0.0002085
— 0M9
— 0 .12
— 0 .12
The reduction of the time is added, only that it may be seen that it is wholly
insensible.
After this, all the longitudes, both of the planet and of the earth, are to be
reduced to the mean vernal equinox for some epoch, for which we shall adopt
the beginning of the year 1805 ; the nutation being subtracted the precession is
to be added, which, for the three observations, is respectively 11".87, 10".23, 8".86,
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
207
so that — 3".56 is to be added for the first observation, — 5".28 for the second,
— 6^74 for the third.
Lastly the longitudes and latitudes of Juno are to be freed from the aberra-
tion of the fixed stars ; thus it is found by well-known rules, that we must sub-
tract from the longitudes respectively 19''.12, 17".ll, 14".82, but add to the lati-
tudes 0".53, r'.18, 1".75, by which addition the absolute values are diminished,
since south latitudes are considered as negative.
151.
All these reductions being properly applied, we have the correct data of the
problem as follows : —
Times of the observations reduced
to the meridian of Paris . Oct. 5.458644 17.421885 27.393077
Longitudes of Juno, a, a', a" . 354°44'3r.60 352°34'22''.12 35r34'30".01
Latitudes, (^, (f, §" _ 4 59 31 .06 —6 21 55 .07 —7 17 50 .95
Longitudes of the earth, I, I', I" 12 28 27 .76 24 19 49 .05 34 16 9 .65
Logs, of the distances, R, R', R" 9.9996826 9.9980979 9.9969678
Hence the calculations of articles 136, 137, produce the following numbers,
7,7,7
d,d',d"
logarithms of the sines
AD, AD', AD" . .
A'D,A:'D',AD" . .
196"=
0' 8'
.36
18
23 59
.20
9.4991995
232
6 26
.44
241
51 15
.22
2
19 34
.00
8.6083885
logarithms of the sines . .
log sin ^ e' ......
log cos ^ g'
Moreover, according to article 138, we have
log tan/? .... 8.9412494 ra log tan ^" . .
log sin (a"—/') . 9.7332391 w logsin(a — /')
log cos («" — /') . 9.9247904 log cos (a— r)
191° 58' 0".33
32 19 24 .93
9.7281105
213 12 29 .82
234 27 0 .90
7 13 37 .70
9.0996915
8.7995259
9.9991357
190°4r40M7
43 11 42 .05
9.8353631
209 43 7 .47
221 13 57 .87
4 55 46 .19
8.9341440
9.1074080 J^
9.6935181^2
9.9393180
208
DETERMINATIOX OF AN ORBIT FROM
[Book II.
Hence
log (tan /5 cos {a" — V) — tan ^" cos {a — /')) = log Tsm t 8.6786513
logsm(«" — a)=logrcosi^ 8.7423191 «
Hence t = 145° 32' 57".78
!{_|-/ = 337 30 58.11
log ^ . . .
log sin it -|- y')
8.8260683
9.5825441 »^
Lastly
log (tan /? sin {a!' — I) — tan /3" sin (a — /))
\o^Tmv{t^Y)
\o^8
8.2033319 ??
8.4086124 w
9.4075427 w "
9.5050667 ?2
9.5376909 w
9.2928554 w
9.2082723^2
a=-f 0.3543592
log cos 9.9915661 7z
« « 9.9853301 n
whence log tan (d' — (T) 9.7947195
8'—a = 31° 56' ir.81, and therefore (7 = 0° 23' 13".12.
According to article 140 we have
riy — d" = 191° 15'18".85 log sin 9.2904352^2
AD'—d =194 48 30 .62
A"D—r =198 39 33 .17
A'D —d'-}-a = 200 10 14 .63
AD'' — d =191 19 8 .27
A[D"—d' + (T = 189 17 46 .06
Hence follow,
loga . . . 9.5494437;
log J . . . 9.8613533.
Formula 13 would give log h = 9.8613531, but we have preferred the former
value, because sin {AD — d' -)- o) is greater than sin {AD" — d' -{-o).
Again, by article 141 we have,
3 log i?' sin ^' . . . 9.1786252
log 2 0.3010300
log sin a 7.8295601
2.6907847
7.3092163 and therefore log c
log 5 9.8613533
loffcoscr 9.9999901
9.8613632
Sect. 1.] theee complete observations. 209
whence = 0.7267135. Hence are derived
d = — 1.3625052, log e = 8.3929518 n
Finally, by means of formulas, article 148, are obtained,
logjt . . . . 0.0913394 ?2
log;t" .... 0.5418957 w
logil . . . . 0.4864480 w
logr .... 0.1592352 w
152.
The preUminary calculations being despatched in this way, we pass to the
first hypothesis. The interval of time (not corrected) between the second and
third observations is 9.971192 days, between the first and second is 11.963241.
The logarithms of these numbers are 0.9987471, and 1.0778489, whence
log ^ = 9.2343285, log ^" = 9.3134303.
"We will put, therefore, for the first ht/pothedsy
:?;=: log P= 0.0791018
^== log ^=8.5477588
Hence we have P = 1.1997804, P + a = 1.5541396, P-\-d= — 0.1627248 ;
logg . . . 8.3929518 w
log(P + a). 0.1914900
C.log{F^d) 0.7885463 w
log tan 0) . . 9.3729881, whence w = + 13°16'5r.89, w + 0 = -f 13°40' 5".01.
log^ . . . 8.5477588
logc . . . 2.6907847
log sin CO . . 9.3612147
log ^c sin w . 0.5997582
The equation
^c sin w sin^ ^= sin (^ — 13°40' 5".01)
is found after a few trials to be satisfied by the value s=: 14°35' 4".90, whence
we have log sin s = 9.4010744, log / = 0.3251340. That equation admits of three
other solutions besides this, namely,
27
210 DETERISimATION OF AN ORBIT FROM [BoOK 11.
s = 32° 2' 28"
0=137 27 59
0 = 193 4 18
The third must be rejected because sui z is negative ; the second because z is
greater than d' ; the first answers to an approximation to the orbit of the earth
of which we have spoken in article 142.
Further, we have, according to article 143,
log^^ 9.8648551
log (P -fa) 0.1914900
C. log sin (s — a). . . . 0.6103678
log'^ 0.6667029
logP 0.0791018
log^ 0.5876011
^-^A'D — d' = z ^199° ^r r.51 =: 214" 22' 6".41; log sin = 9.7516736 ?z
z^A'D"—d'=z-{-lSS 54 32 .94 = 203 29 37 .84; log sin = 9.6005923 w
Hence we have log^= 9.9270735 n, log /'= 0.0226469^, and then
log q = 0.2930977 n, log q" = 0.2580086 n,
whence result
C = 203° 17' 3r.22 log r = 0.3300178
r=110 10 58 .88 logr"= 0.3212819
Lastly, by means of article 144, we obtain
^u"-\-u)= 205° 18' 10".53
^(i/'_m)= — 3 14 2 .02
/'= 3 48 14 .66
log sin 2/ . . . 9.1218791 log sin 2/' . . . 9.1218791
logr 0.3300178 log/' 0.3212819
Clog— .... 9.3332971 C.log^ .... 9.4123989
log sin 2/ . . . 8.7851940 lo^ si^ 2/" . . . 8^8^555599
2/= 3°29"46'.03 2/"= 4°6'43".28
The sum 2/ + 2/" differs in this case from 2/ only by 0".01.
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
211
Now, in order tliat the times may be corrected for aberration, it is necessary to
compute the distances q, q', q" by the formulas of article 145, and afterwards to
multiply them by the time 493*, or 0*^.005706. The following is the calculation,
logr" .... 0.32128
logsin(^''i>'—r) 9.61384
Clog sin r . . 0.16464
logr . . . . 0.33002
logsin(^Z>'— C) 9.23606
C.loffsin^ . . 0.50080
log/
0.32513
log sin [d' —
■^)
9.48384
Clog
sind"'
0.27189
log^ •
log; const.
0.06688
7.75633
log)
0.08086
7.75633
q" . . . . 0.09976
7.75633
log of reduction 7.82321
7.83719
7.85609
reduction =
0.006656
0.006874
0.007179
Observations.
rnn-ected times.
Intervals.
Logarithms.
I.
n.
in.
Oct. 5.451988
17.415011
27.385898
11^.963023
9.970887
1.0778409
0.9987339
The corrected logarithms of the quantities ^, ^", are consequently 9.2343153 and
9.3134223. By commencing now the determination of the elements from /, /,
r", & we obtain log tj =^ 0.0002285, and in the same manner from /", r, /, ^"we
get log tj'' = 0.0003191. "We need not add here this calculation explained at
length in section III. of the first book.
Finally we have, by article 146,
logr .... 9.3134223
C.log^ . . . . 0.7656847
log 7? .... 0.0002285
C.logV' . . . 9.9996809
logP' .... 0.0790164
2 log/ . . .
0.6502680
C. log rr" . .
9.3487003
log^r . . .
8.5477376
QAogriri" . .
9.9994524
Clog cos/ . .
0.0002022
C log cos/' . .
0.0009579
Clog cos/' .
0.0002797
log^'. . . .
8.5475981
The first hypothesis, therefore, results in X = — 0.0000854, Y.
0.0001607.
212
DETEEJtllNATION OF Ali ORBIT FROM
[Book II.
153.
In the second hypothesis we shall assign to P, Q, the very values, which in the
first we have found for P', §'. We shall put, therefore,
a? == log P = 0.0790164
y = log ^ = 8.5475981
Since the calculation is to be conducted in precisely the same manner as in
the first hypothesis, it will be sufficient to set down here its principal results : —
to . , .
log § c sin w
s . . .
loo- / . .
1 n r
log
n' r'
13°15'38'a3
13 38 51 .25
0.5989389
14 33 19 .00
0.3259918
0.6675193
0.5885029
203 16 38 .16
r . .
log r . .
log r"
I {u" -{- u)
h{u" — u)
2/ . .
2/ . .
2r . .
210° 8'24".98
0.3307676
0.3222280
205 22 15 .58
— 3 14 4 .79
7 34 53 .32
3 29 0 .18
4 5 53 .12
It would hardly be worth while to compute anew the reductions of the times
on account of aberration, for they scarcely differ V from those which we have
got in the first hypothesis.
The further calculations furnish log7jz=z 0.0002270, log^" = 0.0003173, whence
are derived
log P'= 0.0790167 X= + 0.0000003
log ^'=8.5476110 F= + 0.0000129
From this it appears how much more exact the second hypothesis is than the
first.
154.
In order to leave nothing to be desired, we will still coUvStruct the third hypothe-
s, in which we shall again choose the values of P', Q', obtained in the second
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
213
hypothesis, as the values of P, Q. Putting, therefore,
:?; = log P = 0.0790167
y = log Q = 8.5476110
the following are found to be the principal results of the calculation : —
oj -\- a .
log Qosino)
log /
log
n r
n'r'
13°15'38'^39
13 38 51 .51
0.5989542
14 33 19 .50
0.3259878
0.6675154
0.5884987
203 16 38 .41
r . .
logr . .
log r"
h {u" -[- u)
h {u" — u)
2/ . .
2/ . .
2/" . .
210° 8'25''.65
0.3307640
0.3222239
205 22 14 .57
— 3 14 4 .78
7 34 53 .73
3 29 0 .39
4 5 53 .34
All these numbers differ so little from those which the second hypothesis fur-
nished, that we may safelj^ conclude that the third hypothesis requires no further
correction.* We may, therefore, proceed to the determination of the elements
from 2/', r, /', &\ which we dispense with transcribing here, since it has already
been given in detail in the example of article 97. Nothing, therefore, remains
but to compute the position of the plane of the orbit by the method of article
149, and to transfer the epoch to the beginning of the year 1805. This computa-
tion is to be based upon the following numbers : —
AD' — i;= 9°55'5r.41
i(/-f«) = 202 18 13 .855
i(;/— m)=— 6 18 5 .495
whence we obtain
l{g^h)= 196°43'14".62
j(^_/^)=:_4 37 24 .41
hi = 6 33 22.05
* If the calculation should be carried through in the same manner as in the preceding hypotheses,
we should obtain X=zO, and Z=-[~^'^0^*^'^0^' which value must be regarded as vanishing, and,
in fact, it hardly exceeds the uncertainty always remaining in the last decimal place.
214 DETERMINATION OF AN OKBIT FROM [BoOK II.
We have, therefore, h = 201° 20' 3r.03, and m9>=l — h = 171° 7' 48".73 ; fur-
ther, g = 192° 5' 50".21, and hence, since the true anomaly for the first oloserva-
tion is found, in article 97, to be 310°55'29".64, the distance of perihelion from
the ascending node in the orbit, 241° 10' 20".57, the longitude of the perihelion
62° 18' 9".30 ; lastly, the inclination of the orbit, 13° 6' 44".10. If we prefer to
proceed to the same calculation from the third place, we have,
A:'D' — 'Q" = 24° 18' 35".25
i (/'+«") =196 24 54 .98
h{7"—u")= — ^ 43 14 .81
Thence are derived
^ (/' J^ h") ^ 211° 24' 32".45
!(/' — //')= — 11 43 48 .48
a = 6 33 22 .05
and hence the longitude of the ascending node, l" — A" = 171° 7'48".72, the lon-
gitude of the perihelion 52° 18' 9".30, the inclination of the orbit 13° 6'44".10,
just the same as before.
The interval of time from the last observation to the beginning of the year
1805 is 64.614102 days; the mean heliocentric motion corresponding to which is
53293".66 =14° 48' 13".66 ; hence the epoch of the mean anomaly at the begin-
ning of the year 1805 for the meridian of Paris is 349° 34' 12".38, and the epoch
of the mean longitude, 41° 52' 21".68.
155.
That it may be more clearly manifest what is the accuracy of the elements
just found, we will compute from them the middle place. For October 17.415011
the mean anomaly is found to be 332° 28' 54".77, hence the true is 315° 1' 23".02
and log r", 0.3259877, (see the examples of articles 13, 14); this true anomaly
ought to be equal to the true anomaly in the first observation increased by the
angle 2/", or to the true anomaly in the third observation diminished by the
angle 2/, that is, equal to 315° 1' 22".98; and the logarithm of the radius vector
should be 0.3259878 : the differences are of no consequence. K the calculation
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
215
for the middle observation is continued to the geocentric place, the results dif-
fer from observation only by a few hundredths of a second, (article 63 ;) these
differences are absorbed, as it were, in the unavoidable errors arising from the
want of strict accuracy in the tables.
"We have worked out the preceding example with the utmost precision, to
show how easily the most exact solution possible can be obtained by our method.
In actual practice it will rarely be necessary to adhere scrupulously to this
type. It will generally be sufficient to use six places of decimals throughout;
and in our example the second hypothesis would have given results not less accu-
rate than the third, and even the first would have been entirely satisfactory. We
imagine that it will not be unacceptable to our readers to have a comparison of
the elements derived from the third hypothesis with those which would result
from the use of the second or first hypothesis for the same object. We exhibit
the three systems of elements in the following table : —
From hypothesis III.
From hypothesis II.
From hypothesis I.
Epoch of mean long. 1805
Mean daily motion . .
Perihelion
Log of semi-axis major .
Ascending node . . .
Inclination of the oi-bit .
41°52'21".68
824".7989
52 18 9 .30
14 12 1 .87
0.4224389
171 7 48 .73
13 6 44 .10
41°52'18".40
824".7983
52 18 6 .66
14 11 59 .94
0.4224392
171 7 49 .15
13 6 45 .12
42°12'37".83
823".5025
52 41 9 .81
14 24 27 .49
0.4228944
171 5 48 .86
13 2 37 .50
By computing the heliocentric place in orbit for the middle observation from
the second system of elements, the error of the logarithm of the radius vector is
found equal to zero, the error of the longitude in orbit, O'^.OS ; and in comput-
ing the same place by the system derived from the first hypothesis, the error of
the logarithm of the radius vector is 0.0000002, the error of the longitude in
orbit, 1".31. And by continuing the calculation to the geocentric place we have,
216
DETERMINATION OF AN ORBIT FROil
[Book 1]
From hypothesis II.
From hypothesis I.
Geocentric longitude
Error
Geocentric latitude .
Error
352° 34' 22".26
0.14
6 21 55 .06
0.01
352° 34' 19".97
2.15
6 21 54.47
0.60
156.
We shall take the second example from Pallas, the following observations of
which, made at Milan, we take from von Zach's Monatliche Correspondens, Vol.
XIV., p. 90.
Mean Time, Milan.
App. Right Ascension.
App. Declination S.
1805, Nov. 5''14'' U'" 4'
Dec. 6 11 51 27
1806, Jan. 15 8 50 36
78° 20' 37".8
73 8 48.8
67 14 11 .1
27° 16' 56".7
32 52 44.3
28 38 8.1
We will here take the equator as the fundamental plane instead of the
ecliptic, and we will make the computation as if the orbit were still wholly un-
known. In the first place we take from the tables of the sun the following data
for the given dates : —
Longitude of the Sun
from mean Equinox.
Distance from
the Earth.
Latitude of
the Sun.
Nov. 5
Dec. 6
Jan. 15
223° 14' 7".61
254 28 42 .59
295 5 47 .62
0.9804311
0.9846753
0.9838153
+ 0".59
+ 0.12
— 0.19
We reduce the longitudes of the sun, the precessions -}-7".59, +3".36, — 2".ll,
being added, to the beginning of the year 1806, and thence we afterwards derive
the right ascensions and declinations, using the mean obliquity 23° 27' 53".53 and
taking account of the latitudes. In this way we find
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
217
Eight ascension of the Sun.
Decl. of the Sun S.
Nov. 5
Dec. 6
Jan. 15
220° 46' 44".65
253 9 23 .26
297 2 51 .11
15° 49' 43".94
22 33 39 .45
21 8 12 .98
These places are referred to the centre of the earth, and are, therefore, .to be
reduced by applying the parallax to the place of observation, since the places of
the planet cannot be freed from parallax. The right ascensions of the zenith to
be used in this calculation agree with the right ascensions of the planet (because
the observations have been made in the meridian), and the declination will be
throughout the altitude of the pole, 45° 28'. Hence are derived the followiag
numbers : —
Eight aso. of the Earth.
Decl. of the Earth N.
Log of dist. from the Sun.
Nov. 5
Dec. 6
Jan. 15
40° 46' 48". 51
73 9 23 .26
117 2 46.09
15° 49' 48".59
22 33 42 .83
21 8 17 .29
9.9958575
9.9933099
9.9929259
The observed places of Pallas are to be freed from nutation and the aberra-
tion of the fixed stars, and afterwards to be reduced, by applying the precession,
to the beginning of the year 1806. On these accoijnts it will be necessary to
apply the following corrections to the observed places : —
28
218
DETERMNATION OF AN ORBIT FROM
[Book IT.
Hence we have the following places of Pallas, for the basis of the compu-
tation : —
Mean Time, Paris.
Eight Ascension.
DecHnation.
Nov. 5.574074
36.475035
76.349444
78° 20' 12".24
73 8 16 .16
67 13 40 .93
— 27° 17' 9".05
— 32 52 48.96
— 28 38 2.42
157.
Now in the first place we will determine the positions of the great circles
drawn from the heliocentric places of the earth to the geocentric places of the
planet. We take the symbols % St', %", for the intersections of these circles
with the equator, or, if you please, for their ascending nodes, and we denote the
distances of the points B, B' , B" from the former points by A, A', A". In the
greater part of the work 'it will be necessary to substitute the symbols 2t, §(', W,
for A, A', A", and also A, A', A" for ^, 8', d" ; but the careful reader will readily
understand when it is necessary to retain A, A', A", d, d', d", even if we fail to
advise him.
The calculation being made, we find
Right ascensions of the
points % r, W . . . 233° 54' 57".10
r, /, f 51 17 15 .74
A, A\ A" 215 58 49 .27
56 26 34 .19
2t'Z>, %D', %D" ... 23 54 52 .13
2l"i>, 5t"i>', %:D" . . 33 3 26 .35
47 1 64 .69
9.8643525
£,!;,£
logarithms of the sines
log sin J e'
log cos i «' ....
253° 8'57".01
90 1 3 .19
212 52 48 .96
55 26 31 .79
30 18 3 .25
31 59 21 .14
89 34 57 .17
9.9999885
9.8478971
9.8510614
276°40'25".87
131 59 58 .03
220 9 12 .96
69 10 57 .84
29 8 43 .32
22 20 6 .91
42 33 41 .17
9.8301910
Sect. 1.] three complete observations. 219
The right ascension of the point W is used in the calculation of article 138
instead of l\ In this manner are found
log I^ sin?! ..... 8.4868236 w
log I' cos ^ 9.2848162 w
Hence t = 189° 2'48".83, log T= 9.2902527 ; moreover, i-{-/ = 279° 3'52".02,
log /S' 9.0110566 n
logTsm{t-\-/) . • . 9.2847950^2
whence J'—a = 208° V 55^^64, and (7 = 4° 50' 53".32.
In the formulas of article 140 sin d, sin d', sin d" must be retained instead of
a, h and -, and also in the formulas of article 142. For these calculations we
have
WD' — J" = 171° 50' 8".18 log sin 9.1523306 log cos 9.9955759^2
%j)'—A =174 19 13.98 « « 8.9954722 « « 9.9978629??
WD— A" =172 54 13.39 « « 9.0917972
<^B —J'-\-o = 175 52 56 .49 « « 8.8561520
^D" — A = 173 9 54 .05 « « 9.0755844
ri/'— ^' + a =174 18 11 .27 « « 8.9967978
Hence we deduce
log;t =0.9211850, logl = 0.0812057 ?2
log x"= 0.8112762, log r= 0.031969172
log a = 0.1099088, az=-\- 1.2879790
log^ =0.1810404,
log- =0.0711314,
whence we have log b = 0.1810402. We shall adopt log 5 = 0.1810403 the
mean between these two nearly equal values. Lastly we have
log c = 1.0450295
d = -\- 0.4489906
log e = 9.2102894
with which the preliminary calculations are completed.
220
DETERMINATION OF AN ORBIT FROM
[Book II.
The interval of time between the second and third observations is 39.874409
days, between the first and second 30.900961 : hence we have
log & = 9.8362757, log&"= 9.7255533.
We put, therefore, for the Jirst hf/jyothesis,
a; = log P=r 9.8892776
y=. log ^=9.5618290
The chief results of the calculation are as follows : —
aj + <7r=20° 8'46".72
log ^c sin w = 0.0282028
Thence the true value of s is 21°ir24''.30, and of log/, 0.3509379. The three
remaining values of s satisfying equation IV., article 141, are, in this instance,
2= 63° 41' 12"
^ = 101 12 58
g=199 24 7
the first of which is to be regarded as an approximation to the orbit of the earth,
the deviation of which, however, is here much greater than in the preceding
example, on account of the too great interval of time. The following numbers
result from the subsequent calculation : —
C . . .
r . .
log r . .
log/' .
^K + ^0
h {u" — u)
2/ . .
2/ . .
2/' ■ .
195° 12' 2".48
196 57 50 .78
0.3647022
0.3355758
266 47 50 .47
-43 39 5 .33
22 32 40 .86
13 5 41 .17
9 27 0 .05
We shall distribute the difference between 2/' and 2/-|-2/", which in this case
is 0".36, between 2/ and 2/" in such a manner as to make 2/= 13° 5'40".96,
and2/"=9°26'59".90.
The times are now to be corrected for aberration, for which purpose we are to
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
221
put in the formulas of article 145,
We have, therefore,
r=ri>'— //"+.
logr . . .
0.36470
log/
. .
. 0.35094
log sin {AD' —
C) 9.76462
log sin ((5'' —
-s) 9.75038
Clog sin d .
0.07918
Clog
sin(^'
. 0.08431
log const. . .
7.75633
7.96483
log const. .
. 7.75633
7.94196
reduction of"!
the time J
0.009222
0.008749
Hence follow.
Observations.
Corrected times.
Intervals.
I.
n.
in.
Nov. 5.564852
36.466286
76.340252
30''.901434
39.873966
log/' .... 0.33557
logsin(yl"X>'—r) 9.84220
Clog sin (^" . . 0.02932
log const. . . 7.75633
7.96342
0.009192
Logarithms.
1.4899785
1.6006894
whence are derived the corrected logarithms of the quantities &, 6" respectively
9.8362708 and 9.7265599. Beginning, then, the calculation of the elements
from /, /', 2/, 6, we get log rj = 0.0031921, just as from r, r, 2f", ^" we obtain
log ri" = 0.0017300. Hence is obtained
log P'= 9.8907512
and, therefore,
X= +0.0014736
The chief results of the second
log $'=9.5712864,
Y= +0.0094574
!5, in which we put
a; = log P = 9.8907512
a^ = log $z= 9.5712864
are the following ;
w + a . .
log ^ c sin w
z . . . .
lose / . . .
20° 8' 0".87
0.0373071
21 12 6 .09
0.3507110
C 195° 16' 59".90
r 196 52 40 .63
logr .... 0.3630642
loff/' . . . . 0.3369708
222
DETERMINATION OF AN ORBIT FROM
[Book H.
i{u''-\-u). . . 267" 6'10''.75 2/ 22° 32' 8".69
^u" — u). . . — 43 39 4.00 2/ 13 154.65
2/' .... . 9 30 14 .38
The difference 0."34, between 2/' and 2/ -|- 2/'" is to be so distributed, as to
make 2/= 13° 1' 54".45, 2f' == 9° 30' 14".24.
If it is thought worth while to recompute here the corrections of the times,
there will be found for the first observation, 0.009169, for the second, 0.008742,
for the third, 0.009236, and thus the corrected times, November 5.564905, Novem-
ber 36.466293, November 76.340280. Hence we have
log^ 9.8362703 I log?/' 0.0017413
log^ 9.7255594 logP' 9.8907268
log?; 0.0031790 | log ^' 9.5710593
Accordingly, the results from the second hypothesis are
X = — 0.0000244, r== — 0.0002271.
Finally, in the third hypothesis, in which we put
a; = log P = 9.8907268
y = log ^ = 9.5710593
the chief results of the calculation are as follows : —
(o-^a . . . . 20° 8' 1".62
log Qc sin (o . . 0.0370857
3 21 12 4 .60
log/ 0.3507191
C 195 16 54 .08
r 196 52 44 .45
logr 0.3630960
The difference 0".38 will be here distributed in such a manner as to make
2f= 13° 1' 57".20, 2/" = 9° 30' 10".47.*
logr" . .
0.3369536
Htc^' + u).
. 267 5 53 .09
Hic"-u).
.—43 39 4.19
2/' . . .
. 22 32 7 .67
2/ . . .
. 13 1 57 .42
2/" . . .
9 30 10 .63
* This somewhat increased difference, nearly equal in all the hypotheses, has arisen chiefly from
this, that (T had been got too little by almost two hundredths of a second, and the logaritlmi of b too
great by several units.
Sect. 1.] three complete observations. 223
Since the differences of all these numbers from those which the second
hypothesis furnished are very small, it may be safely concluded that the third
hypothesis requires no further correction, and, therefore, that a new hypothesis
would be superfluous. Wherefore, it will now be proper to proceed to the calcu-
lation of the elements from 2/', ^', r, / ' : and since the processes comprised in
this calculation have been most fully explained above, it will be sufficient to add
here the resulting elements, for the benefit of those who may wish to perform the
computation themselves : —
Eight ascension of the ascending node on the equator .... 158° 40' 38".93
Inclination of the orbit to the equator 11 42 49 .13
Distance of the perihelion from the ascending node 323 14 56 .92
Mean anomaly for the epoch 1806 335 4 13 .05
Mean daily (sidereal) motion 770".2662
Angle of eccentricity, (p 14 9 3 .91
Logarithm of the semi-axis major . 0.4422438
158.
The two preceding examples have not yet furnished occasion for using the
method of article 120 : for the successive hypotheses converged so rapidly that
we might have stopped at the second, and the third scarcely difiered by a sensible
amount from the truth. We shall always enjoy this advantage, and be able to do
without the fourth hypothesis, when the heliocentric motion is not great and the
three radii vectores are not too unequal, particularly if, in addition to this, the
intervals of the times differ from each other but little. But the further the con-
ditions of the problem depart from these, the more will the first assumed values
of F and Q differ from the correct ones, and the less rapidly will the subsequent
values converge to the truth. In such a case the first three hypotheses are to
be completed in the manner shown in the two preceding examples, (with this
difference only, that the elements themselves are not to be computed in the third
hypothesis, but, exactly as in the first and second hypotheses, the quantities r], rf^
P', Q', X, Y) ; but then, the last values of P', Q' are no longer to be taken as
224
DETERMINATION OF AN ORBIT FROM
[B(
n.
the new values of the quantities P, Q in the new hypothesis, but these are to
be derived from the combination of the first three hypotheses, agreeably to the
method of article 120. It will then very rarely be requisite to proceed to the
fifth hypothesis, according to the precepts of article 121. We will now explain
these calculations further by an example, from which it will appear how far our
method extends.
159.
For the third example we select the following observations of Ceres, the first
of which has been made by Olbers, at Bremen, the second by Haeding, at Got-
tingen, and the third by Bessel, at Lilienthal.
Mean time of place of observation.
Bight Ascension.
North declination.
1805, Sept. 5^3* 8^54'
1806, .Jan. 17 10 58 51
1806, May 23 10 23 53
95° 59' 25"
101 18 40.6
121 56 7
22° 21' 25"
30 21 22.3
28 2 45
As the methods by which the parallax and aberration are taken account of,
when the distances from the earth are regarded as wholly unknown, have already
been sufficiently explained in the two preceding examples, we shall dispense
with this unnecessary increase of labor in this third example, and with that
object will take the approxunate distances from von Zach's Monaillclie Correr
spondenz, Vol. XL, p. 284, in order to free the observations from the effects of
parallax and aberration. The following table shows these distances, together
with the reductions derived from them : —
Distance of Ceres from the earth . . .
Time in which the light reaches the earth
Eeduced time of observation
Sidereal time in degrees
Parallax in right ascension
Parallax in declination
2.899
1.638
2.964
23'«49'
13'"28-'
24'" 21'
2H5'" 5'
10H5'"23^
9"59'"32
355° 55'
97° 59'
210° 41
+ 1".90
-f 0".22
— 1".97
— 2.08
— 1.90
— 2.04
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
225
Accordingly; the data of the problem, after being freed from parallax and
aberration, and after the times have been reduced to the meridian of Paris, are as
follows : —
Times of the observations.
Eight Ascension.
DecUnation.
1805, Sept. 5, 12^ 19™ 14'
1806, Jan. 17, 10 15 2
1806, May 23, 9 33 18
95° 59' 23".10
101 18 40.38
121 56 8.97
22° 21' 27".08
30 21 24.20
28 2 47.04
From these right ascensions and declinations have been deduced the longi-
tudes and latitudes, using for the obliquity of the ecliptic 23° 27' 55''.90, 23° 2T
54".59, 23° 27' 63".27 ; the longitudes have been afterwards freed from nutation,
which was for the respective times -|- 17".31, -|- 17''.88, -(- 18".00, and next re-
duced to the beginning of the year 1806, by applying the precession -\- 15''.98,
— 2''.39, — 19".68. Lastly, the places of the sun for the reduced times have
been taken from the tables, in which the nutation has been omitted in the longi-
tudes, but the precession has been added in the same way as to the longitudes of
Ceres. The latitude of the sun has been wholly neglected. In this manner have
resulted the following numbers to be used in the calculation : —
Times, 1805, September
or, a', a"
/5,i3',r- .....
l,V,l"
log R, log R, log R' .
5.51336
95° 32' 18".56
— 0 59 34 .06
342 54 56 .00
0.0031514
139.42711
99° 49' 5".87
4-7 16 36 .80
117 12 43 .25
9.9929861
265.39813
118° 5'28".85
+ 7 38 49 .39
241 58 50 .71
0.0056974
The preliminary computations explained in articles 136-140 furnish the fol-
io win o; : —
r, r , r
AD, AD', AD"
A'D, A'D', AD'
358°55'28".09
112 37 9 .66
15 32 41 .40
138 45 4 .60
29 18 8 .21
29
156°52'11".49
18 48 39 .81
252 42 19 .14
6 26 41 .10
170 32 59 .08
170°48'44".79
123 32 52 .13
136 2 22 .38
358 5 57 .00
156 6 25 .25
226
DETERMINATION OF AN ORBIT FROM
[Book IL
log e = 0.8568244
log K = 0.1611012
logj'/'= 9.9770819 ?z
log 2. = 9.9164090 w
loo;r= 9.7320127 w
a = 8° 52' 4".05
log a = 0.1840193 n,a= — 1.5276340
log i = 0.0040987
log c= 2.0066735
f?= 117.50873
The interval of time between the first and second observations is 133.91375
days, between the second and third, 125.97102 : hence
log ^ = 0.3358520, log r=: 0.3624066, log -^'=0.0265546, log ^r = 0.6982586.
We now exhibit in the following table the principal results of the first three
hypotheses : —
I
n.
ni.
logP^a;
0.0265546
0.0256968
0.0256275
logQ = ?/
0.6982586
0.7390190
0.7481055
w-fa
7°15'13".523
7°14'47".139
7°14'45".071
log ^c sin to
1.1546650 ?z
1.1973925 ;2
1.2066327 ^z
e
7 3 59 .018
7 2 32 .870
7 2 16 .900
log/
0.4114726
0.4129371
0.4132107
c
160 10 46 .74
160 20 7 .82
160 22 9 .42
r
262 6 1 .03
262 12 18 .26
262 14 19 .49
logr
0.4323934
0.4291773
0.4284841
log r^
0.4094712
0.4071975
0.4064697
*K+^0
262 55 23 .22
262 57 6 .83
262 57 31 .17
n«"— "/
273 28 50 .95
273 29 15 .06
273 29 19 .56
2/
62 34 28 .40
62 49 56 .60
62 53 57 .06
2/
31 8 30 .03
31 15 59 .09
31 18 13 .83
2r
31 25 58 .43
31 33 57 .32
31 35 43 .32
logrj
0.0202496
0.0203158
0.0203494
logV'
0.0211074
0.0212429
0.0212751
logP'
0.0256968
0.0256275
0.0256289
log Q'
0.7390190
0.7481055
0.7502337
X
— 0.0008578
— 0.0000693
-f 0.0000014
Y
4-0.0407604
+ 0.0090865
4- 0.0021282
Sect. 1.]
THREE COMPLETE OBSERVATIONS.
227
If we designate the three values of X by A, A', A''; the three values of Y by
B, B\ B"; the quotients arising from the division of the quantities A'B" — A!'B',
A^'B — AB", AB' — AB, by. the sum of these quantities, by Ji, Jc\ k", respectively,
so that we have ^-j-^'-|-£'=: 1 ; and, finally, the values of log P' and log Q' in the
third hj^othesis, by M and N, (which would become new values of x and p if it
should be expedient to derive the fourth hypothesis from the third, as the third
had been derived from the second) : it is easily ascertained from the formulas of
article 120, that the corrected value of x is M — Jc [A -\- A') — k'A', and the cor-
rected value of y, iV^ — k [B' -f- B") — k'B". The calculation being made, the
former becomes 0.0256331, the latter, 0.7509143. Upon these corrected values
we construct the fourth hypothesis^ the chief results of which are the following : —
CO -{- (7 . .
log ^csinw
0 . . . .
log / . . .
^ . . . .
7°14'45".247
1.2094284 n
7 2 12 .736
0.4132817
160 22 45 .38
log /'
h{%l' — u)
2/ . .
2/ . .
2/' • .
0.4062033
262°5r38".78
273 29 20 .73
62 55 16 .64
31 19 1 .49
31 36 15 .20
r 262 15 3 .90
^ logr 0.4282792
The difference between 2/' and 2/-|- 2/" proves to be 0''.05, which we shall
distribute in such a manner as to make 2/= 31° 19' r.47, 2/'= 31° 36' 15".17.
If now the elements are determined from the two extreme places, the following
values result : —
True anomaly for the first place 289° 7'39".75
True anomaly for the third place 352 2 56 .39
Mean anomaly for the first place 297 41 35 .65
Mean anomaly for the third place 353 15 22 .49
Mean daily sidereal motion 769^.6755
Mean anomaly for the beginning of the year 1806 . . 322 35 52 .51
Angle of eccentricity, 9 4 37 57 .78
Logarithm of the semi-axis major 0.4424661
By computing from these elements the heliocentric place for the time of the
228 DETERMINATION OF AN ORBIT FROM [BoOK 11.
middle observation, the mean anomaly is found to be 326° 19' 25".72, the loga-
rithm of the radius vector, 0.4132825, the true anomaly, 320° 43' 54".87 : this last
should differ from the true anomaly for the first place by the quantity 2/", or
from the true anomaly for the third place by the quantity 2/, and should, there-
fore, be 320° 43' 54".92, as also the logarithm of the radius vector, 0.4132817 :
the difference 0".05 in the true anomaly, and of eight units in the logarithm, is
to be considered as of no consequence.
If the fourth hypothesis should be conducted to the end in the same way as
the three preceding, we would have X= 0, Y= 0.0000168, whence the follow-
ing corrected values of x and y would be obtained,
X = log P = 0.0256331, (the same as in the fourth hypothesis,)
^ = log ^ = 0.7508917.
If the fifth hypothesis should be constructed on these values, the solution would
reach the utmost precision the tables allow: but the resulting elements would
not differ sensibly from those which the fourth hypothesis has furnished.
Nothing remains now, to obtain the complete elements, except that the posi-
tion of the plane of the orbit should be computed. By the precepts of article
149 we have
From the first place. From the third place.
ff 354° 9' 44".22 / . . . . 57° 5' 0".91
h 261 56 6 .94 r .... 161 0 1 .61
i 10 37 33 .02 10 37 33 .00
9, 80 58 49 .06 80 58 49 .10
Distance of the perihelion
,, 65 2 4 .47 65 2 4 .52
from the ascendmg node
Longitude of the perihelion 146 0 53 .53 146 0 53 .62
The mean being taken, we shall put i= 10° 37' 33".01, 9, = 80° 58' 49".08, the
longitude of the perihelion = 146° 0' 53".57. Lastly, the mean longitude for
the beginning of the year 1806 will be 108° 36' 46".08.
Sect. 1.] three complete observations. 229
160.
In the exposition of the method to which the preceding investigations have
been devoted, we have come upon certain special cases to which it did not apply,
at least not in the form in which it has been exhibited by us. We have seen
that this defect occurs first, when any one of the three geocentric places coincides
either with the corresponding heliocentric place of the earth, or with the oppo-
site point (the last case can evidently only happen when the heavenly body
passes between the sun and earth) : second, when the first geocentric place of the
heavenly body coincides with the third j third, when aU three of the geocentric
places together with the second heliocentric place of the earth are situated in the
same great circle.
In the first case the position of one of the great circles AB, AB', A.'B", and in
the second and third the place of the point B^, wUl remain indeterminate. In
these cases, therefore, the methods before explained, by means of which we have
shown how to determine the heliocentric from the geocentric places, if the quan-
tities P, Q, are regarded as known, lose their efl&cacy : but an essential distinction
is here to be noted, which is, that in the first case the defect will be attributable
to the method alone, but in the second and third cases to the nature of the prob-
lem; in the first case, accordingly, that determination can undoubtedly be effected
if the method is suitably altered, but ia the second and third it will be absolutely
impossible, and the heliocentric places will remain indeterminate. It will not be
uninteresting to develop these relations in a few words : but it would be out of
place to go through all that belongs to this subject, the more so, because in all
these special cases the exact determination of the orbit is impossible where it
would be greatly afiected by the smallest errors of observation. The same defect
will also exist when the observations resemble, not exactly indeed, but nearly,
any one of these cases ; for which reason, in selecting observations this is to be
recollected, and properly guarded against, that no place be chosen where the
heavenly body is at the same time in the vicinity of the node and of opposition
or conjunction, nor such observations as where the heavenly body has nearly re-
turned in the last to the geocentric place of the first observation, nor, finally, such
230 DETEEMINATION OF AN ORBIT FROM [BoOK 11.
as where the great circle drawn from the middle heliocentric place of the earth to
the middle geocentric place of the heavenly body makes a very acute angle with
the direction of the geocentric motion, and nearly passes through the first and
third places.
161.
We will make three subdivisions of the first case.
I If the point B coincides with A or with the opposite point, 8 will be equal
to zero, or to 180° ; y, t', t" and the points D', D", will be indeterminate ; on the
other hand, y', y", e and the points B, B% will, be determinate ; the point C will
necessarily coincide with A. By a course of reasoning similar to that pursued in
article 140, the following equation will be easily obtained : —
^ , sin (z — a) R sin d' sin {A!'D — 8") „
^ ~ ^^ sin ^ i2" sin ,5" sin {A!D — 5' + <t) '^ '
It will be proper, therefore, to apply in this place all which has been explained in
articles 141, 142, if, only, we put a = 0, and h is determined by equation 12,
article 140, and the quantities z, r', — , — ^, will be computed in the same manner
as before. Now as soon as z and the position of the point C have become
known, it will be possible to assign the position of the great circle CC, its inter-
section with the great circle A'B", that is the point G", and hence the arcs CC,
CC", CC", or 2/", 2f', 2/. Lastly, from these will be had
n'r' sin 2 f „ ?iVsin 2 f"
n sin 2/' ' vl' sin 2/' '
n. Every thing we have just said can be applied to that case in which B"
coincides with A!.' or with the opposite point, if, only, all that refers to the first
place is exchanged with what relates to the third place.
HE. But it is necessary to treat a little difierently the case in which B' coin-
cides with A. or with the opposite point. There the point C will coincide with
A. ', Y, e, t" and the points D,D",B^, will be indeterminate: on the other hand,
the intersection of the great circle BB" with the ecliptic,f the longitude of which
t More generally, with the great circle AA' : but for the sake of brevity we are now considering
that case only where the ecliptic is taken as the fundamental plane.
Sect. 1.] three complete observations. 231
maybe put equal to l' -\- tt, may be determined. By reasonings analogous to
those which have been developed in article 140, will be obtained the equation
S sin d sin (A" ly — 8")
E" sin d" sin (AD' — d) ' R" sin {I" -- V — n) '
Let us designate the coefficient of n, which agrees with a, article 140, by the
same symbol a, and the coefficient of n'r by /i : a may be here also determined
by the formula
Rsin{l'-[-7t — l)
^~ R"sin{l" — l'—ny
We have, therefore,
0 = aJ^-f-/3^^y-{-w",
which equation combined with these,
produces
whence we shall be able to get /, unless, indeed, we should have /3 = 0, in which
case nothing else would follow from it except P = — a. Further, although we
might not have (? = 0 (when we should have the third case to be considered in
the following article), still (3 will always be a very small quantity, and therefore
P wi|l necessarily differ but little from — a : hence it is evident that the deter-
mination of the coefficient
is very uncertain, and that /, therefore, is not determinable with any accuracy.
Moreover, we shall have
wV t^i^ ^ P+a
~^~ ~f~' n" ~~ ~JP~'
after this, the following equations will be easily developed in the same manner as
in article 143,
r sm C = — ^-r sm (r — I),
232 DETERMINATION OF AN ORBIT FROM [BoOK 11.
// • ^11 n'/sinr . ,,, ,,
r" ^Vixi,' ^— ^^r^mx{l —I),
r sin U: — AD') = r"P '^^ sin (l," — A'D'),
from the combination of which with equations VIII. and IX. of article 143, the
quantities r, t? '>'"■) Q" can be determined. The remaining processes of the calcula-
tion will agree with those previously described.
162.
In the second case, where B" coincides with B, D' will also coincide with them
or with the opposite point. Accordingly, we shall have AD' — d and A'l/ — d"
either equal to 0 or 180° : whence, from the equations of article 143, we obtain
sms' R sin (5
= +
sin i' H' sin 5"
n" — sin a" sin (z -|- A' I/' — «5') '
R sin d sin e" sin {z -\- AD" — d') = PR" sin d" sin e sin (s + A'D — d').
Hence it is evident that z is determinable by P alone, independently of Q, (un-
less it should happen that AD" = AD, or = AD + 180°, when we should have
the third case) : s being found, / will also be known, and hence, by means of
the values of the quantities
and, lastly, from this also
n'r' nV ^ n ■, n"
— , —ff, also -r and -7;
^ = 2fc + 'i;-l)/l
Evidently, therefore, P and Q cannot be considered as data independent of each
other, but they will either supply a single datum only, or inconsistent data. The
positions of the points C, 0" will in this case remain arbitrary, if they are only
taken in the same great circle as 0' .
In the third case, where A, B, B', B", lie in the same great circle, D and D" will
coincide with the points B", B, respectively, or with the opposite points : hence is
Sect. 1.] THREE COMPLETE OBSERVATIONS. 233
obtained from the combination of equations VII., VIII., IX., article 143,
p ^ sin 5 sin &" B sin (I' — I)
i^" sin fi" sine R"sm{l" — V) '
In this case, therefore, the value of P is had from the data of the problem, and,
therefore, the positions of the points 0, C, G", will remain indeterminate.
163.
The method which we have fully explained from article 136 forwards, is prin-
cipally suited to the first determination of a wholly unknown orbit : still it is em-
ployed with equally great success, where the object is the correction of an orbit
already approximately known by means of three observations however distant
from each other. But in such a case it will be convenient to change some things.
When, for example, the observations embrace a very great heliocentric motion, it
will no longer be admissible to consider -j and & ^" as approximate values of the
quantities P, Q : but much more exact values will be obtained from the very
nearly known elements. Accordingly, the heliocentric places in orbit for the
three times of observation will be computed roughly by means of these elements,
whence, denoting the true anomalies by v, v' , v", the radii vectores by r, /, r'\ the
semi-parameter by j», the following approximate values will result : —
p r sin {v' — v) ^ 4 r'^ sin \ (v' — v) sin \ {v" — v')
/' sin (vl' — w') ' ^ pcos^ (v" — v)
With these, therefore, the first hypothesis will be constructed, and with them, a
little changed at pleasure, the second and third : it would be of no advantage
to adopt F' and Q' for the new values, since we are uo longer at liberty to sup-
pose that these values come out more exact. For this reason all three of the
hypotheses can be most conveniently despatched at the same time: the fourth will
then be formed according to the precepts of article 120. FinaiUy, we shall not
object, if any person thinks that some one of the ten methods explained in arti-
cles 124-129 is, if not more, at least almost equally expeditious, and prefers to
use it,
30
SECOND SECTION.
DETERSIINATION OF AN ORBIT FROM FOUR OBSERVATIONS, OF WHICH TWO
ONLY ARE COMPLETE.
164.
We have already, in the beginning of the second hook (article 115), stated
that the use of the problem treated at length in the preceding section is lim-
ited to those orbits of which the inclination is neither nothing, nor very small,
and that the determination of orbits slightly inclined must necessarily be based
on four observations. But four complete observations, since they are equivalent
to eight equations, and the number of the unknown quantities amounts only to
six, would render the problem more than determinate : on which account it wiU
be necessary to set aside from two observations the latitudes (or declinations),
that the remaining data may be exactly satisfied. Thus a problem arises to
which this section will be devoted : but the solution we shall here give will ex-
tend not only to orbits slightly inclined, bul; can be applied also with equal suc-
cess to orbits, of any inclination however great. Here also, as in the problem of
the preceding section, it is necessary to separate the case, in which the approxi-
mate dimensions of the orbit are already known, from the first determination
of a wholly unknown orbit : we will begin with the former.
165.
The simplest method of adjusting a known orbit to satisfj^ four observations
appears to be this. Let x^ y, be the approximate distances of the heavenly body
from the earth in two complete observations : by means of these the correspond-
ing heliocentric places may be computed, and hence the elements; after this,
(234)
Sect. 2.] determination of an orbit. 235
from these elements the geocentric longitudes or right ascensions for the two
remaining observations may be computed. If these happen to agree with the
observations, the elements will require no further correction: but if not, the
differences X, Y, will be noted, and the same calculation will be repeated twice,
the values of x, y being a little changed. Thus will be obtained three systems
of values of the quantities x, y, and of the differences X, Y, whence, according
to the precepts of article 120, will be obtained the corrected values of the quan-
tities :?;, ^, to which will correspond the values X== 0, Jr=: 0. From a similar
calculation based on this fourth system elements will be found, by which all four
observations will be correctly represented.
If it is in your power to choose, it will be best to retain those observations
complete from which the situation of the orbit can be determined with the great-
est precision, therefore the two extreme observations, when they embrac-e a helio-
centric motion of 90° or less. But if they do not possess equal accuracy, you
will set aside the latitudes or declinations of those you may suspect to be the
less accurate.
166.
Such places will necessarily be used for the first determination of an entirely
unknown orbit from four observations, as include a hehocentric motion not too
great ; for otherwise we should be without the aids for forming conveniently the
first approximation. The method which we shall give directly admits of such
extensive application, that observations comprehending a heliocentric motion of
30° or 40° may be used without hesitation, provided, only, the distances from the
sun are not too unequal : where there is a choice, it will be best to take the
intervals of the times between the first and second, the second and third, the
third and fourth but little removed from equality. But it will not be necessary
to be very particular in regard to this, as ,the annexed example wUl show, in
which the intervals of the times are 48, 55, and 59 days, and the heliocentric
motion more than 50°.
Moreover, our solution requires that the second and third observations be
complete, and, therefore, the latitudes or declinations in the extreme observations
236 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BoOK II.
are neglected. We have, indeed, shown above that, for the sake of accuracy, it is
generally better that the elements be adapted to two extreme complete observa-
tions, and to the longitudes or right ascensions of the intermediate ones ; never-
theless, we shall not regret having lost this advantage in the first detennination
of the orbit, because the most rapid approximation is by far the most important,
and the loss, which affects chiefly the longitude of the node and the inclina-
tion of the orbit, and hardly, in a sensible degree, the other elements, can after-
wards easily be remedied.
We will, for the sake of brevity, so arrange the explanation of the method,
as to refer all the places to the ecliptic, and, therefore, we will suppose four longi-
tudes and two latitudes to be given : but yet, as we take into account the latitude
of the earth in our formulas, they can easily be transferred to the case in which
the equator is taken as the fundamental plane, provided that right ascensions and
declinations are substituted in the place of longitudes and latitudes.
Finally, all that we have stated in the preceding section with respect to nutar
tion, precession, and parallax, and also aberration, applies as well here : unless,
therefore, the approximate distances from the earth are otherwise known, so that
method I., article 118, can be employed, the observed places will in the beginning
be freed from the aberration of the fixed stars only, and the times will be cor-
rected as soon as the approximate determination of the distances is obtained in
the course of the calculation, as will appear more clearly in the sequel.
167.
We preface the explanation of the solution with a list of the principal sym-
bols. We will make
i, i', f, f, the times of the four observations,
a, a, a", a'", the geocentric longitudes of the heavenly body,
/i, /?', /5", (i"\ their latitudes,
r, r', /', r", the distances from the sun,
q, ()', q", q'", the distances from the earth,
I, I', I", I'", the heliocentric longitudes of the earth,
Sect. 2.] OF which two only are complete. 237
B, B', B", B'", the heliocentric latitudes of the earth,
R, R', R'\ R'", the distances of the earth from the sun,
(wOl), (w 12), («23), (w 02), (« 13), the duplicate areas of the triangles which
are contained between the sun and the first and second places of the heavenly
body, the second and third, the third and fourth, the first and third, the second
and fourth respectively; [ri 01), [r] 12), (tj 23) the quotients arising from the
division of the areas h {n 01), 2 {n 12), ^ [n 23), by the areas of the correspond-
ing sectors ;
(n 01) ' ~ (n 23) '
V, v', v", v'", the longitudes of the heavenly body in orbit reckoned from an arbi-
trary point. Lastly, for the second and third observations, we will denote the
heliocentric places of the earth in the celestial sphere by JL', JL", the geocentric
places of the heavenly body by B\ B", and its heliocentric places by C, C" .
These things being understood^ the first step will consist, exactly as in the
problem of the preceding section (article 136), in the determination of the posi-
tions of the great circles A!. C'B', A' C"B", the inclinations of which to the eclip-
tic we denote by /', y": the determination of the arcs AB'=id', A'B"z=d" will be
connected at the same time with this calculation. Hence we shall evidently have
r' = sl {q'q' + 2 q'R' cos d' -\- R'R')
r"= si {q'Y + 2 Q"Ji" cos 6" + R"R''),
or by putting 9' + R' cos d' = x', q" + R" cos d" = a!', R! sin d' — a', R" sin d" = a",
r' = \J (// + a'a')
r"=sl{x"^'-\-a;'a").
168.
By combining equations 1 and 2, article 112, the following equations in sjm.-
bols of the present discussion are produced : —
0 = («12)i?cos^sin(/— a) — (?^02)(9'cos/5'sin(a'— a)-f i2'cos^'sin(r— a))
4- {n 01) (y" cos {i" sin {a" — «) + R" cos B" sin {I" — a)),
238 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BuOK II.
0 = {n 23) (^/ COS /5' sin (a'"— «') + B' cos B' sin (a'"— /'))
— {n 13) (9" cos /?"sin (a"'— a") + i^'' cos B'' sin (a'"— T))
+ {n 12) i?"' cos B'" sin (a'" — r").
These equations, bj putting
i2' cos 5' sin (Z' — a) -r,/ .>/ 7/
n, . , , r-^ B COS a =J,
cos p sin (a — a) '
jB"cos5"sm(a'"— Z") „„ ^„ ,„
-coT^'sMi?^^-^ ««^^ =^'
ig'cos^sin(«--0_ ^ ^^^ ^, _ ^,
cos (3 sm (a — a ) '
jy- cos ^;- sin (I" -a) _ ^. ^^^ ^. ^ ^.
cosp sin (« — a)
i? cos B sin (Z — a) .
cos (3" sin (a" — «) '
Jg^"cos^"Mn(«^^^ — r^O _ . ,„
cos/3'sin («'" — «') ~ »
cos §' sin (a' — a) /
cos ^' sin (a" — a) ^'
cos|y^sin(«'^^ — «'0 „
cos |3'sin («'" — a') ^ '
and all the reductions being properly made, are transformed into the following: —
/(l-f-PO(:c^ + y)
14
Q'
+ x" + ^i^,
iH — -3
(a/'s/'-\-a"a"f
or, by putting besides,
_ k'— rF"= c", ^" (1 + P") = /',
into these,
L y' = o-+ ^' <-" + ''>
(xV -}-«'«')
Sect. 2.] OF wmcH two only ake complete. 239
(a/V'+a"a")*
With the aid of these two equations / and x' can be determined from d, h', c', d',
Q'. a", h", c", d", Q". If, indeed, / or x" should be eliminated from them, we should
obtain an equation of a very high order : but still the values of the unknown
quantities x', x", will be deduced quickly enough from these equations by indi-
rect methods without any change of form. Generally approximate values of
the unknown quantities result if, at first, Q' and Q" are neglected ; thus : —
^ _ </^ -f d" {W -I- c') -f d'd"b'
^— 1—d'd" '
„_c'-\- d' {V + </0 + d'd"V'
^ — 1-d'd"
But as soon as the approximate value of either unknown quantity is obtained,
values exactly satisfying the equations will be very easily found. Let, for ex-
ample, t' be an approximate value of x, which being substituted in equation I.,
there results x" =-1" ', in the same manner from x" = ^" being substituted in
equation II., we may have x' =^ X'; the same processes may be repeated by sub-
stituting for X in I., another value h,' -j- v', which may give x" = ^" -\- v" ; this
value being substituted in II., may give x' = X' -\- N\ Thereupon the corrected
value of X will be
5 "T iV'—y — i^'_/
and the corrected value of x",
If it is thought worth while, the same processes will be repeated with the cor-
rected value of x' and another one sHghtly changed, until values of x', x" satisfy-
ing the equations I., 11. exactly, shall have been found. Besides, means will not
be wanting even to the moderately versed analyst of abridging the calculation.
In these operations the irrational quantities (a;V-|- aV)^', (a;' V-f-«"fif")^ ^^®
conveniently calculated by introducing the arcs /, z", of which the tangents are
240 DETERMINATION OF AN ORBIT FROM FOUK OBSERVATIONS, [BoOK II.
respectively ^, -7,, whence come
v/(^V-[-«V)=/ = ^ = ^,
' ^ ' ' sm/ cosz"
I / r/ r/ \ r/ n\ it O, X
\J ix X 4-aa ) = r =:-^ — 7, = j,.
' ^ ' ' sin 2" cosz'
These auxiliary arcs, which must be taken between 0° and 180°, in order that
/, r", may come out positive will, manifestly, be identical with the arcs C'B', G"B",
whence it is evident that in this way not only / and r", but also the situation of
the points C, C", are known.
This determination of the quantities x', x" requires a', a", V , h", c\ c", d', d'\ (^,
Q" to be known, the first four of which quantities are, in fact, had from the data
of the problem, but the four following depend on P', P". Now the quantities
P', P", Q', Q", cannot yet be exactly determined ; but yet, since
^- ^ -t'-t (rjuy
TV P" ^'~^' C'?^^)
V. Q =hJcJc{if — t){t — f) ^, (^ 01) {t] 12) cos 1 (v' — v) cos i (t/' — v) cos \ (i-" — v') '
VI. Q" = i kic {f— /) (r — f) y^ P2) (v?23) cos\{v"—v') cosi (z;'"-«') cos\^{v"' — v")'
the approximate values are immediately at hand,
p' ^ ^ jp" ^' — ^
q = hJck it' — t) [f — If), Q" =HJc {t" — t') {f — r ),
on which the first calculation will be based.
169.
The calculation of the preceding article being completed, it will be necessary
first to determine the arc C C". Which may be most conveniently done, if, as
in article 137, the intersection D of the great circles AfC'B', Al' C"B", and their
mutual inclination e shall have been previously determined: after this, will be
found from £, CD = s'-\- B'D, and 0"J) = z" -\- B"D, by the same formulas
Sect. 2.] or which two only are complete. 241
which we have given in article 144, not only C G" ^v" — /, but also the angles
{%{, u",) at which the great circles AB', A'B", cut the great circle C C".
After the arc v" — v' has been found, v' — y, and r will be obtained from a
combination of the equations
r sm [y — v)^= ^^t ,
r sin {v' — v -^ v" — /) — , p, „,
-I I V
1+.
and ia the same manner, /" and v'" — v" from a combination of these: —
in • r III II \ 1^ sin (v' — v')
r sm (y — V ) = —^ ,
r'" sin {v'" - v" + v" - v') = ^^- ^'^-<^'-f) .
1 + J
All the numbers found in this manner would be accurate if we could set out in
the beginning from true values of P', Q^, P", Q'^ : and then the position of the
plane of the orbit might be determined in the same manner as in article 149,
either from A'C, u' and /, or from A' C", u" and y" -, and the dimensions of the
orbit either from /, /', t', f, and v" — v', or, which is more exact, from r, r", t,
f, v'" — V. But in the first calculation we will pass by all these things, and will
direct our attention chiefly to obtaining the most approximate values of P' , P" .
Q', Q". We shall reach this end, if by the method explained in 88 and the fol-
lowing articles,
from r, /, v' — v,f — t we obtain {rj 01)
« /^r",v" — v',f — t' « (7jl2)
« r"yy—v",f'—f •'' (7j23).
We shall substitute these quantities, and also the values of r, /, /', /", cos i {v' — v),
etc., in formulas III.- VI., whence the values of P', Q', P", Q" will result much
more exact than those on which the first hypothesis had been constructed. With
these, accordingly, the second hypothesis will be formed, which, if it is carried to
a conclusion exactly in the same manner as the first, will furnish much more
exact values of I*', Q', P", Q', and thus lead to the third hypothesis. These
processes will continue to be repeated, until the values of P', Q, P", Q" seem to
31
242 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BoOK II.
require no further correction, how to judge correctly of which, frequent practice
will in time show. When the heliocentric motion is small, the first hypothesis
generally supplies those values with sufficient accuracy : but if the motion in-
cludes a greater arc, if, moreover, the intervals of the times are very unequal,
hypotheses several times repeated will be wanted ; but in such a case the first
hypotheses do not demand great preciseness of calculation. Finally, in the last
hypothesis, the elements themselves will be determined as we have just indicated.
170.
It will be necessary in the first hypothesis to make use of the times t, if, f, f,
uncorrected, because the distances from the earth cannot yet be computed': as
soon, however, as the approximate values of the quantities x', x" have become
known, we shall be able to determine also those distances approximately. But
yet, since the formulas for q and q'" come out here a little more complicated, it
will be weU to put off the computation of the correction of the times until the
values of the distances have become correct enough to render a repetition of the
work unnecessary. On which account it will be expedient to base this operation
on those values of the quantities x', x", to which the last hypothesis but one leads,
so that the last hypothesis may start with corrected values of the times and of
the quantities P', P'\ Q'y Q". The following are the formulas to be employed
for this purpose : —
Vn. q' = x'—EQOBd\
Ym. f=xf'—R'coBd",
IX. Qcos(i = — i? cos -5 cos (a — /)
-j — — ^, (q' cos (^' cos (a' — a) -\- E' cos B' cos U' — a))
— -^(^" cos ^" COS («"—«) 4-^" COS ^" cos (r—«)),
X. Qsm^ = — JRsmB^ ^^"^-^ ((>' sin (r + R' sin B")
^'(1 + 7-a)
— -i (^/' sin (r + B" sin B") ,
Sect. 2.] OF which two only aee complete. 243
XI. q'" cos r = — R'" cos B'" cos {a'" —I'")
H ^^'^" rv' (q" cos ir COS {a'" — a") -{- R" cos B" cos (a'" — l"))
xn.
— 'mW cos ii' cos {a'" — a) -\- R' cos B' cos (a'" — I'yj,
r sin /3"' = — R!" sin ^"' -\ ^"^^^, (^'' sin ^" + iT' sin .g'')
^"(l + ls)
^,(9'sin/'r-hi?'sin^').
The formulas IX.-XII. are derived without difficulty from equations 1, 2, 3, article
112, if, merely, the symbols there used are properly converted into those we here
employ. The formulas will evidently come out much more simple if B, B' , B"
vanish. Not only (), but also /5 will follow from the combination of the formulas
IX. and X., and, in the same manner, besides r" , also {^"' from XI. and XII. : the
values of these, compared with the observed latitudes (not entering into the
calculation), if they have been given, will show with what degree of accuracy
the extreme latitudes may be represented by elements adapted to the six remain-
ing data.
171.
A suitable example for the illustration of this mvestigation is taken from Vesta,
which, of all the most recently discovered planets, has the least inclination to
the ecliptic* We select the following observations made at Bremen, Paris,
Lilienthal, and Milan, by the illustrious astronomers Olbers, Bouvard, Bessel, and
Oriani : —
* Nevertheless this inclination is still great enough to admit of a sufficiently safe and accurate deter-
mination of the orbit based upon three observations : in fact the first elements which had been derived
in this way from observations only 19 days distant from each other (see von Zach's Monatliche Gor-
respondenz, Vol. XV. p. 595), approach nearly to those which were here deduced from four observa-
tions, removed from each other 162 days.
244
DETERlNnNATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BoOK II.
Mean time of place of observation.
Riglit Ascension.
Declination.
1807, March 30, 12'^ 33"* 17*
May 17, 8 16 5
July 11, 10 30 19
Sept. 8, 7 22 16
183° 52' 40".8
178 36 42.3
189 49 7.7
212 50 3.4
11° 54' 27".0 N.
11 39 46.8
3 9 10 .IN.
8 38 17 .OS.
We find for the same times from the tables of the sun,
Longitude of the Sun
from app. Equinox.
Nutation.
Distance from
the Earth.
Latitude of
the Sun.
Apparent obliquity
of the Ecliptic.
March 30
May 17
July 11
Sept. 8
9° 21' 59".5
55 56 20 .0
108 34 53 .3
165 8 57 .1
:
hl6.8
-16.2
-17.3
-16.7
0.9996448
1.0119789
1.0165795
1.0067421
+ 0".23
— 0.63
— 0.46
+ 0.29
23° 27' 50".82
49.83
49 .19
23 27 49 .26
The observed places of the planets have, the apparent obliquity of the ecHp-
tic being used, been converted into longitudes and latitudes, been freed from
nutation and aberration of the fixed stars, and, lastly, reduced, the precession
being subtracted, to the beginning of the year 1807 ; the fictitious places of the
earth have then been derived from the places of the sun by the precepts of arti-
cle 72 (in order to take account of the parallax), and the longitudes transferred
to the same epoch by subtracting the nutation and precession ; finally, the times
have been counted from the beginning of the year and reduced to the meridian
of Paris. In this manner have been obtained the following; numbers : —
i,t',t",r . .
89.505162
, 137.344502
192.419502
251.288102
a, a', a", a'". .
178° 43' 38".87
174° r30".08
187°45'42".23
213°34'15".63
(i,(r,(r,r^ .
12 27 6 .16
10 8 7 .80
6 47 25 .51
4 20 21 .63
I, r, r, r . .
189 21 33 .71
235 56 0 .63
288 35 20 .32
345 9 18 .69
log B,R',R",R"'
9.9997990
0.0051376
0.0071739
0.0030625
Hence we deduce
/=168°32'4r.34, r= 62° 23' 4".8
8, log a = 9.9526104,
/ = 173
5 15 .68, d"
= 100 45 1 .4
0, loga'=9
.9994839,
Sect. 2.] of which two only ake complete. 245
y = — 11.009449, x' = — 1.083306, log I = 0.0728800, log / = 9.7139702 n
b" = — 2.082036, x'' = -1-6.322006, logr'= 0.0798512?z log/*" =9.8387061
AD= 37°ir5r.50, rD= 89°24'ir.84, £ = 9° 5' 5".48
B'D = — 26 5 13 .38, B"I) = — 11 20 49 .56.
These preliminary calculations completed, we enter upon the ^rst hypothesis.
From the intervals of the times we obtain
log i{i; — t) = 9.9153666
logy&(r — 0 = 9.9765359
logy^(r— 0 = 0.0054651,
and hence the first approximate values
log P' = 0.06117, log (1 -f F') = 0.33269, log Q" = 9.59087
logP"=: 9.97107, log(l -f-P") = 0.28681, log Q"= 9.67997,
hence, further,
e' = — 7.68361, log d' = 0.04666 n
c"= -\- 2.20771, log d"= 0.12552.
With these values the following solution of equations I., IE., is obtained, after a
few trials : —
x' = 2.04856, / = 23° 38' 17", log / = 0.34951
/'= 1.95745, 0"=:27 2 0, logr"= 0.34194.
From s', z" and £, we get
C"C"=«;" — y' = 17° r 5":
hence v' — v, r, v'" — y", r", will be determinable by the following equations : —
log r sin {v' — v)= 9.74942, log r sin (/ — y -j- 17° 1' 5") = 0.07600
log/"sin(y"'—z;")= 9.84729, log/"sin(/"— y''^- 17 7 5") = 0.10733
whence we derive
v' — v = 14° 14' 32", log r = 0.35865
v"'—v"= 18 48 33, logr"'= 0.33887.
Lastly, is found
log {n 01) = 0.00426, log {n 12) == 0.00599, log {n 23) = 0.00711,
and hence the corrected values of P', P", Q', Q\
246 DETER]\nNATION OF AN ORBIT FROM FOUR OBSERYATIONS, [BoOK 11.
log P' = 0.05944, log Q' = 9.60374,
log P"= 9.97219, log Q"= 9.69581,
upon which the second hjpoihem will be constructed. The principal results of this
are as follows : —
c' = — 7.67820, log ^' = 0.045736 n
c"= + 2.21061, logc/"= 0.126054
d = 2.03308, / = 23° 47' 54", log / = 0.346747,
:?;"= 1.94290, g''= 27 12 25, log /'== 0.339373
C'C"=^v" — v'=\T 8' 0''
v' — v= 14° 21' 36", log r = 0.354687
v"'—v"=\'^ 50 43, logr"'= 0.334564
log {n 01) = 0.004359, log (w 12) = 0.006102, log [n 23) = 0.007280.
Hence result newly corrected values of P' , P", Q', Q",
log P' = 0.059426, log Q' = 9.604749
log P" = 9.972249, log Q" = 9.697564,
from which, if we proceed to the third hypothesis, the follomng numbers result : —
d =. — 7.67815, log d! = 0.045729 n
c" = 4- 2.21076, log d"= 0.126082
x' = 2.03255, z = 23° 48' 14", log / = 0.346653
^"=1.94235, /'=27 12 49, log r"= 0.339276
C'0"=v''—v'=l7° 8' 4"
v' — v= 14° 21' 49", logr =0.354522
t,'"_z;''=18 51 7, log/"= 0.334290
log (w 01) = 0.004363, log (^? 12) = 0.006106, log 0? 23) = 0.007290.
If now the distances from the earth are computed according to the precepts of
the preceding article, there appears : — •
9' = 1.5635, 9" =2.1319
log 9 cos /3 = 0.09876 log q" cos /?'" = 0.42842
log 9 sin /? = 9.44252 log 9'" sin |3"' = 9.30905
/? = 12° 26' 40" /5'" = 4° 20' 39"
log 9 = 0.10909 log 9'" = 0.42967.
Sect. 2.]
OF WHICH TWO ONLY ARE COMPLETE.
247
Hence are found
Corrections of the Times.
Corrected Times.
I.
0.007335
89.497827
n.
0.008921
135.335581
m.
0.012165
192.407337
IV.
0.015346
251.272756
whence wiU result newly corrected values of the quantities P', JP", Q', Q",
log P' == 0.059415, log Q' = 9.604782,
logP''= 9.972253, log Q'' = 9.697687.
Finally, if the fourth hypothesk is formed with these new values, the following
numbers are obtained : —
c' = — 7.678116, log d! = 0.045728
c"= + 2.210773, log ^"=0.126084
x' = 2.032473, s' = 23° 48' 16". 7, log / = 0.346638
a;"= 1.942281, ^'=27 12 51.7, log /'= 0.339263
v''—v' = 17° 8' 5".l, i(«t''4-w0 = 176° r^0".6, ^u"—u') = ^°SS'2r.e
v' — v = U 21 51 .9, log r = 0.354503
v'''—v"=lS 51 9 .5, logr'"= 0.334263
These numbers differ so little from those which the third hypothesis furnished,
that we may now safely proceed to the determination of the elements. In
the first place we get out the position of the plane of the orbit. The inclina-
tion of the orbit 7° 8' 14".8 is found by the precepts of article 149 from /', iif,
and ^'^' = ^' — /, also the longitude of the ascending node 103° 16'37".2, the
argument of the latitude in the second observation 94° 36' 4". 9, and, there-
fore, the longitude in orbit 197° 52' 42".l ; in the same manner, from 7", it", and
-A" C" = d" — /', are derived the inclination of the orbit = 7° 8' 14".8, the longi-
tude of the ascending node 103° 16' 37".5, the argument of the latitude in the
third observation 111° 44' 9". 7, and therefore the longitude in orbit 215° 0'47".2.
Hence the longitude in orbit for the first observation will be 183° 30' 50".2, for
the fourth 233° 51' 56". 7. If now the dimensions of the orbit are determined
from r — t, r, r'", and v'" — y = 50° 21' 6".5, we shall have.
248 DETERJiONATION OF AN ORBIT FROM FOUR OBSERVATIONS. [BoOK 11.
True anomaly for the first place 293°33'43".7
True anomaly for the fourth place 343 54 50 .2
Hence the longitude of the perihelion 249 57 6 .5
Mean anomaly for the first place 302 33 32 .6
Mean anomaly for the fourth place 346 32 25.2
Mean daily sidereal motion 9 7 8''. 72 16
Mean anomaly for the beginning of the year 1807 . 278 13 39 .1
Mean longitude for the same epoch 168 10 45 .6
Angle of eccentricity (p 5 2 58 .1
Logarithm of the semi-axis major 0.372898
K the geocentric places of the planet are computed from these elements
for the corrected times t, f, f, f, the four longitudes agree with «, a', a'\ a", and
the two intermediate latitudes with {^', 1^", to the tenth of a second ; but the
extreme latitudes come out 12° 26' 43".7 and 4° 20' 40".l. The former in error
22".4 in defect, the latter 18".5 in excess. But yet, if the inclination of the
orbit is only increased 6", and the longitude of the node is diminished 4' 40", the
other elements remaining the same, the errors distributed among all the latitudes
will be reduced to a few seconds, and the longitudes will only be affected by the
smallest errors, which will themselves be almost reduced to nothing, if, in addition.
2" is taken from the epoch of the longitude.
THIKD SECTION.
THE DETERMINATION OF AN ORBIT SATISFYING AS NEARLY AS POSSIBLE ANY
NUMBER OF OBSERVATIONS WHATEVER.
172.
If the astronomical observations and other quantities, on which the computa-
tion of orbits is based, were absolutely correct, the elements also, whether deduced
from three or four observations, would be strictly accurate (so far indeed as the
motion is supposed to take place exactly according to the laws of Keplee), and,
therefore, if other observations were used, they might be confirmed, but not cor-
rected. But since all our measurements and observations are nothing more than
approximations to the truth, the same must be true of all calculations resting
upon them, and the highest aim of all computations made concerning concrete
phenomena must be to approximate, as nearly as practicable, to the truth. But
this can be accomplished in no other way than by a suitable combination of
more observations than the number absolutely requisite for the determination of
the unknown quantities. This problem can only be properly undertaken when
an approximate knowledge of the orbit has been already attained, which is after-
wards to be corrected so as to satisfy all the observations in the most accurate
manner possible.
It then can only be worth while to aim at the highest accuracy, when the
final correction is to be given to the orbit to be determined. But as long as it
appears probable that new observations will give rise to new corrections, it will
be convenient to relax more or less, as the case may be, from extreme precision,
if in this way the length of the computations can be considerably diminished.
We win endeavor to meet both cases.
32 (249)
250 DETERMINATION OF AN ORBIT FROM [BoOK II.
173.
Ill the first place, it is of the greatest importance, that the several positions of
the heavenly body on which it is proposed to base the orbit, should not be
taken from single observations, but, if possible, from several so combined that the
accidental errors might, as far as may be, mutually destroy each other. Obser-
vations, for example, such as are distant from each other by an interval of a few
days, — or by so much, in some cases, as an interval of fifteen or twenty days, —
are not to be used in the calculation as so many different positions, but it would
be better to derive from them a single place, wliich would be, as it were, a mean
among all, admitting, therefore, much greater accuracy than single observations
considered separately. This process is based on the following principles.
The geocentric places of a heavenly body computed from approximate ele-
ments ought to differ very little from the true places, and the differences between
the former and latter should change very slowly, so that for an interval of a
few days they can be regarded as nearly constant, or, at least, the changes may
be regarded as proportional to the times. If, accordingly, the observations should
be regarded as free from all error, the differences between the observed places
corresponding to the times t, t', if', if", and those which have been computed from
the elements, that is, the differences between the observed and the computed
longitudes and latitudes, or right ascensions and declinations, would be quanti-
ties either sensibly equal, or, at least, uniformly and very slowly increasing or de-
creasing. Let, for example, the observed right ascensions a, a', a", a", etc., cor-
respond to those times, and let a -\-d, a' -\- d', a" -\- d'\ a" -\- d'", etc., be the
computed ones ; then the differences d, d', d", d'", etc. will differ from the true
deviations of the •elements so far only as the observations themselves are errone-
ous : if, therefore, these deviations can be regarded as constant for aU these ob-
servations, the quantities d, d', d", d'", etc. will furnish as many different determi-
nations of the same quantity, for the correct value of which it will be proper to
take the arithmetical mean between those determinations, so far, of course, as
there is no reason for preferring one to the other. But if it seems that the same
degree of accuracy cannot be attributed to the several observations, let us assume
Sect. 3.]' any number of observations. 251
that the degree of acc-uracj in each may be considered proportional to the num-
bers e, /, e", /", etc. respectively, that is, that errors reciprocally proportional to
these numbers could have been made in the observations with equal facility;
then, according to the principles to be propounded below, the most probable
mean value will no longer be the simple arithmetical mean, but
e e g -f ^e'8' -[- d'd'^' + e"'e"'b"' + etc.
ee-\- e'e! -f- eV -|- e"'e"' -(-etc.
Putting now this mean value equal to A, we can assume for the true right ascen-
sions, a -[- (^ — A,a! -\- cf — J, a' \ 8" — ■ A, a!"\- 8'" — A, respectively, and then
it will be arbitrary, which we use in the calculation. But if either the observa-
tions are distant from each other by too great an interval of time, or if suffi-
ciently approximate elements of the orbit are not yet known, so that it would
not be admissible to regard their deviations as constant for all the observations, it
will readily be perceived, that no other difference arises from this except that the
mean deviation thus found cannot be regarded as common to all the observa-
tions, but is to be referred to some intermediate time, which must be derived from
the individual times in the same manner as A from the corresponding deviations,
and therefore generally to the time
eetA^ e'e't' -{- e"e"t" -A^ e"'e"'t"' -|- etc.
e e -}- e'e' -\- e"e" -\-e"'e"' -\-<iic. '
Consequently, if we desire the greatest accuracy, it will be necessary to compute
the geocentric place from the elements for the same time, and afterwards to free
it from the mean error A, in order that the most accurate position may be ob-
tained. But it will in general be abundantly sufficient if the mean error is
referred to the observation nearest to the mean time. What we have said -here
of right ascensions, applies equally to declinations, or, if it is desired, to longitudes
and latitudes : however, it will always be better to compare the right ascensions
and declinations computed from the elements immediately with those observed ;
for thus we not only gain a much more expeditious calculation, especially if we
make use of the methods explained in articles 53-60, but this method has the
additional advantage, that the incomplete observations can also be made use of;
and besides, if every thing should be referred to longitudes and' latitudes, there
252 DETERMINATION OF AN ORBIT FROM [BoOK II
would be cause to fear lest an observation made correctly in right ascension,
but badly in declination (or the opposite), should be vitiated in respect to both
longitude and latitude, and thus become wholly useless. The degree of precision
to be assigned to the mean found as above will be, according to the principles to
be explained hereafter,
Sl{ee-^ e'e' -{- e"e" -{- e"'e"' + etc.) ;
so that four or nine equally exact observations are required, if the mean is to
possess a double or triple accuracy.
174.
If the orbit of a heavenly body has been determined according to the methods
given in the preceding sections from three or four geocentric positions, each one
of which has been derived, according to the precepts of the preceding article,
from a great many observations, that orbit will hold a mean, as it were, among
all these observations ; and in the differences between the observed and computed
places there will remain no trace of any law, which it would be possible to re-
move or sensibly diminish by a correction of the elements. Now, when the whole
number of observations does not embrace too great an interval of time, the best
agreement of the elements with all the observations can be obtained, if only
three or four normal positions are judiciously selected. How much advantage
we shall derive from this method in determining the orbits of new planets or
comets, the observations of which do not yet embrace a period of more than
one year, will depend on the nature of the case. When, accordingly, the orbit
to be determined is inclined at a considerable angle to the ecliptic, it will be
m general based upon three observations, which we shall take as remote from
each other as possible : but if in this way we should meet with any one of the
cases excluded above (articles 160-162), or if the inclination of the orbit should
seem too small, we shall prefer the determination from four positions, which, also,
we shall take as remote as possible from each other.
But when we have a longer series of observations, embracing several years,
more normal positions can be derived from them ; on which account, we should
Sect. 3.] any number of observations. 253
not insure the greatest accuracy, if we were to select three or four positions only
for the determination of the orbit, and neglect all the rest. But in such a case,
if it is proposed to aim at the greatest precision, we shall take care to collect
and employ the greatest possible number of accurate places. Then, of course,
more data will exist than are required for the determination of the unknown
quantities : but all these data will be liable to errors, however small, so that it
will generally be impossible to satisfy all perfectly. Now as no reason exists,
why, from among those data, we should consider any six as absolutely exact, but
since we must assume, rather, upon the principles of probability, that greater or
less errors are equally possible in all, promiscuously ; since, moreover, generally
speaking, small errors oftener occur than large ones ; it is evident, that an orbit
which, while it satisfies precisely the six data, deviates more or less from the
others, must be regarded as less consistent with the principles of the calculus of
probabilities, than one which, at the same time that it differs a little from those
six data, presents so much the better an agreement with the rest. The investigar
tion of an orbit having, strictly speaking, the maximum probability, will depend
upon a knowledge of the law according to which the probability of errors de-
creases as the errors increase in magnitude: but that depends upon so many
vague and doubtful considerations — physiological included — which cannot be
subjected to calculation, that it is scarcely, and indeed less than scarcely, possible
to assign properly a law of this kind in any case of practical astronomy. Never-
theless, an investigation of the connection between this law and the most prob-
able orbit, which we will undertake in its utmost generality, is not to be regarded
as by any means a barren speculation.
175.
To this end let us leave our special problem, and enter upon a very general
discussion and one of the most fruitful in every application of the calculus to
natural philosophy. Let V, V, V", etc. be functions of the unknown quantities
j», q, r. s, etc., ^ the nmnber of those functions, v the number of the unknown
quantities ; and let us suppose that the values of the functions found by direct
observation are V = M, V = M', V" = M", etc. Generally speaking, the
254 DETERmXATIOX OF AN ORBIT FROM [BoOK 11.
determination of the unknown quantities will constitute a problem, indetermi-
nate, determinate, or more than determinate, according as ii<iv, /^ = v, or
/,< > v:'' We shall confine ourselves here to the last case, in which, evidently, an
exact representation of all the observations would only be possible when they
were all absolutely free from error. And since this cannot, in the nature of
things, happen, every system of values of the unknown quantities p, q, r, s, etc.,
must be regarded as possible, which gives the values of the functions V — M,
V — M', V" — M", etc., within the limits of the possible errors of observation ;
this, however, is not to be understood to imply that each one of these systems
would possess an equal degree of probability.
Let us suppose, in the first place, the state of things in all the observations to
have been such, that there is no reason why we should suspect one to be less
exact than another, or that we are bound to regard errors of the same magnitude
as equally probable in all. Accordingly, the probability to be assigned to each
error J will be expressed by a function of J which we shall denote by ^> A. Now
although we cannot precisely assign the form of this function, we can at least
af&rm that its value should be a maximum for ^ = 0, equal, generally, for equal
opposite values of A, and should vanish, if, for A is taken the greatest error, or a
value greater than the greatest error: 9)//, therefore, would appropriately be re-
ferred to the class of discontinuous functions, and if we undertake to sub.'^titute
any analytical function in the place of it for practical purposes, this must be of
such a form that it may converge to zero on both sides, asymptotically, as it were,
from // = 0, so that beyond this limit it can be regarded as actually vanishing.
Moreover, the probability that an error lies between the limits A and A -\- AA
differing from each other by the infinitely small difference d z/, will be expressed
byy^dz/; hence the probability generally, that the error lies between i> and
* If, in the third case, the functions V, V, V" should be of such a nature that ju -j- 1 — 1' of them,
or more, might be regarded as functions of the remainder, the problem would still be more than determi-
nate with respect to these functions, but indeterminate with respect to the quantities p, q, r, s, etc. ; that
is to say, it would be impossible to determine the values of the latter, even if the values of the func-
tions V, V, V", etc. should be given with absolute exactness : but we shall exclude this case from our
discussion.
Sect. 3.] any numbee of obseryations. 255
i)', will be given by the integral / 9) //.dz/ extended from J = D to J=^D'.
This integral taken from the greatest negative value of z/ to the greatest positive
value, or more generally from z/ = — co to z/ = -[-<» must necessarily be equal
to unity. Supposing, therefore, any determinate system of the values of the
quantities p, q, r, s, etc., the probability that observation would give for V the
value M, will be expressed by 9 {M — V), substituting in V for p, q, r, s, etc.,
their values; in the same manner cp {M' — F), (f [M" — V"), etc. will express the
probabilities that observation would give the values M', M", etc. of the func-
tions V, V", etc. Wherefore, since we are authorized to regard all the observa-
tions as events independent of each other, the product
^ ^M—V) (f [M'—V) if {M"—V") etc., =i2
will express the expectation or probability that all those values will result to-
gether from observation. -
176.
Now in the same manner as, when any determinate values whatever of the
unknown quantities being taken, a determinate probability corresponds, previ-
ous to observation, to any system of values of the functions F, V, V", etc. ; so,
inversely, after determinate values of the functions have resulted from observa-
tion, a determinate probability will belong to every system of values of the un-
known quantities, from which the values of the functions could possibly have
resulted : for, evidently, those systems will be regarded as the more probable in
which the greater expectation had existed of the event which actually occurred.
The estimation of this probability rests upon the following theorem : —
If, any hjpoiliesis H leing made, the probability of any determinate event E is h, and
if, another hypothesis H' being made excluding the former and equally probahle in itself, the
probability of the same event is h' ; then I say, tvhen the event E has actually occurred, thai
the probability that H tvas the true hypothesis, is to the probability that H' was the true
hypothesis, as h to \x.
For demonstrating which let us suppose that, by a classification of all the cir-
cumstances on which it depends whether, with H or H' or some other hypothesis,
256
DETERMINATION OF AN ORBIT FROM
[Book II.
the event E or some other event; should occur, a system of the different cases is
formed, each one of which cases is to be considered as equally probable in itself
(that is, as long as it is uncertain whether the event E, or some other, will occur),
and that these cases be so distributed.
that among them
may be found
in which should be assumed
the hypothesis
in such a mode as would give
occasion to the event.
m
H
E
n
H
different from E
m'
H'
E
n'
H'
different from E
rri'
different from H and H'
E
7i'
different from ^and H'
different from E
Then we shall have
h =
',' m -\-n '
moreover, before the event was known the probability of the hypothesis H was
m-\-n
m -^ n -\- n^ -\- n' -\- rd' -\- ri' '
but after the event is known, when the cases n, n', if disappear from the number
of the possible cases, the probability of the same hypothesis will be
7n-\-m -\-m '
in the same way the probability of the hypothesis H' before and after the event,
respectively, will be expressed by
.'JrW
and
m -\- n -\- m! -\- n' -\- ??/' -|- Ji" " " m-\- m' -\-
since, therefore, the same probability is assumed for the hypotheses R and H'
before the event is known, we shall have
m -\-n = m^ -\- n',
whence the truth of the theorem is readily inferred.
Now, so far as we suppose that no other data exist for the determination of
the unknown quantities besides the observations V=M, V' = M', V":^M",
Sect. 3.] a^y number of observations. 257
etc., and, therefore, that all systems of values of these unknown quantities were
equally probable previous to the observations, the probability, evidently, of any
determinate system subsequent to the observations will be proportional to S2.
This is to be understood to mean that the probability that the values of the un-
known quantities lie between the infinitely near limits j» and jo-j-djo, q and g-\-dq,
r and r-j- dr, s and 5 -|- ^^j ^tc. respectively, is expressed by
Xi2dj!?d^drd5 , etc.,
where the quantity 1 will be a constant quantity independent of p, q, r, s, etc. :
and, indeed, ^ will, evidently, be the value of the integral of the order v,
f^£ldipdiqdrds , etc.,
for each of the variables p, q, r, 5, etc., extended from the value — 00 to the
value -[- GO •
177.
Now it readily follows from this, that the most probable system of values of
the quantities p, q, r, s, etc. is that in which il acquu-es the maximum value, and,
therefore, is to be derived from the v equations
:^ = 0, -— = 0, -5—= 0, -7- = 0, etc.
These equations, by putting
V—M=:v, r—M' = v', V"—M" = v", etc., and ^^, = ^>' ^,
assume the following form : —
d^9^ + d^9^+a^9^ +etc.= 0,
r^^'^ + d^^'^+d^?'^ +etc.= 0,
Av f . Av' , , . Avl' , ,f . . „
■^r'f ^^J-rf^ +d79^^ -l-etc.= 0,
i ■
Av , . Av' , , , Av" r rr \ , a
dl9'^ + d79'^+d79^^ +etc.= 0.
Hence, accordingly, a completely determinate solution of the problem can be
obtained by elimination, as soon as the nature of the function 9' is known. Since
33
258 DETERJMINATION OF AN ORBIT FROM [BoOK 11.
this cannot be defined a priori, we will, approaching the subject from another
point of view, inqnh-e upon what function, tacitly, as it were, assumed as a
base, the common principle, the excellence of which is generally acknowledged,
depends. It has been customary certainly to regard as an axiom the hypothesis
that if any quantity has been determined by several direct observations, made
under the same circumstances and with equal care, the arithmetical mean of the
observed values affords the most probable value, if not rigorously, yet very
nearly at least, so that it is always most safe to adhere to it. By putting,
therefore,
F=F'=F"etc.=j9,
we ought to have in general,
9' {M—p) -f 9' {M' —p) -f ^>' [M" — p)-\- etc. = 0,
if instead of p is substituted the value
-(ilf-f ilf' + Jf"+etc.),
wnatever positive integer }i expresses. By supposing, therefore,
M' = M''= etc. =M—iiN,
we shall have in general, that is, for any positive integral value of \i,
whence it is readily inferred that ^ must be a constant quantity, which we will
denote by k. Hence we have
log (^ A^=\'kA A -\- Constant,
denoting the base of the hyperbolic logarithms by e and assuming
Constant = log x.
Moreover, it is readily perceived that I" must be negative, in order that S2 may
really become a maximum, for which reason we shall put
and since, by the elegant theorem first discovered by Laplace, the integral
Sect. 3.] any number of observations. 259
from J :^ — oo to// = -|-oo is^, (denoting bj n the semicircumference of
the circle the radius of which is unity), our function becomes
178.
The function just found cannot, it is true, express rigorously the probabilities
of the errors: for since the possible errors are in all cases confined within certain
limits, the probability of errors exceeding those limits ought always to be zero,
while our formula always gives some value. However, this defect, which every
analytical function must, from its nature, labor under, is of no importance in
practice, because the value of our function decreases so rapidly, when hJ has
acquired a considerable magnitude, that it can safely be considered as vanishing.
Besides, the nature of the subject never admits of assigning with absolute rigor
the limits of error.
Finally, the constant h can be considered as the measure of precision of the
observations. For if the probability of the error J is supposed to be expressed
in any one system of observations by
_^ .-hhAA
and in another system of observations more or less exact by
the expectation, that the error of any observation in the former system is con-
tained between the limits — d and -|- d will be expressed by the integral
/
^^-hhA^^J
taken from J = — dto J z=-^ d ; and in the same manner the expectation, that
the error of any observation in the latter system does not exceed the limits — d'
and -|- d' will be expressed by the integral
I
^g-h-h'AA^JJ
extended from J = — d' to J = -\-d' : but both integrals manifestly become
260 DETERMDsATION OF AN ORBIT FROM [BoOK II.
equal when we have /^ cT = h'd'. Now, therefore, if for example // = 2 A, a double
error can be committed in the former system with the same facility as a single
error in the latter, in which case, according to the common way of speaking, a
doable degree of precision is attributed to the latter observations.
179.
We will now develop the conclusions which follow from this law. It is evi-
dent, in order that the product
may become a maximum, that the sum
vv -\- v'v' -\- v"v" -j- etc.,
must become a minimum. Therefore, tJiat will he the most prolable syste^n of values of
the unknown quantities p, q, r, s, etc., in which the sum of the squares of the diff^et-emes
between the observed and computed values of the functions V, V, Y", etc. is a minimum, if
the same degree of accuracy is to be presumed in all the observations. This prin-
ciple, which promises to be of most frequent use in all applications of the mathe-
matics to natural philosophy, must, everywhere, be considered an axiom with
the same propriety as the arithmetical mean of several observed values of the
same quantity is adopted as the most probable value.
This principle can be extended without difficulty to observations of unequal
accuracy. K, for example, the measures of precision of the observations by
means of which V=^M, V = M', V" =iM", etc. have been found, are expressed,
respectively, by h, h', h", etc., that is, if it is assumed that errors reciprocally pro-
portional to these quantities might have been made with equal facility in those
observations, this, evidently, will be the same as if, by means of observations of
equal precision (the measure of which is equal to unity), the values of the func-
tions hV, h'V, h"V", etc., had been directly found to be hM, // Jf' , A"ilf ", etc. :
wherefore, the most probable system of values of the quantities 2h §"? '"? •^j ^tc,
will be that in which the sum of hhvv -|- liliv'v' -\- li'H'v'v" -\- etc , that is, in tvhich
the sum of the squares of the differences between the actuall// observed and computed values
)nuUipllcd by numbers that measure the degree of precision, is a minimum. In this way it
Sect. 3.] any nibiber of observations. 261
is not even necessary that the functions V, V, V", etc. relate to homogeneous
quantities, but thej may represent heterogeneous quantities also, (for example,
seconds of arc and time), provided only that the ratio of the errors, which might
have been committed with equal faciHty in each, can be estimated.
180.
The principle explained in the preceding article derives value also from this,
that the numerical determination of the unknown quantities is reduced to a very
expeditious algorithm, when the functions V, V, V'\ etc. are linear. Let us
suppose
V — Jtf = v=^ — m -\- a2} -\- h q -\- cr -\- ds -\- etc.
V — M' = y' = — m' -\- a'p -\- h'q -j- cr -j- d's -\- etc.
V" — M"^=v"=^ — m"-\- a"p -|- h"q -j- c"r -\- d"s -\- etc.
etc., and let us put
av -\- a'v' -\- a"v" -\- etc. = P
hv -\- b'v' -f b"v" + etc. = Q
ev -\- c'v' -j- c"v" -|- etc. ^ R
dv + d'v' + d"v" + etc. = >S'
etc. Then the v equations of article 177, from which the values of the unknown
quantities must be determined, will, evidently, be the following : —
P = 0, Q=^,R=^%S=^, etc.,
provided we suppose the observations equally good ; to which case we have shown
in the preceding article how to reduce the others. We have, therefore, as many
linear equations as there are unknown quantities to be determined, from which
the values of the latter will be obtained by common elimination.
Let us see now, whether this elimination is always possible, or whether the
solution can become indeterminate, or even impossible. It is known, from the
theory of elimination, that the second or third case will occur when one of the
equations
P=% Q = 0,B=0, S=0, etc.,
being omitted, an equation can be formed from the rest, either identical with the
262 DETERMINATION OF AN OllBIT FKOM [BoOK II.
omitted one or inconsistent with it, or, which amounts to the same thing, when
it is possible to assign a linear function
aP^i^Q^yR-y dS + etc.,
which is identically either equal to zero, or, at least, free from all the unknown
quantities /», q, r, s, etc. Let us assume, therefore,
aP-\-(SQ-]-yB-\-dS-{- etc. = ;c.
We at once have the identical equation
[v -\-m)v-\- {v' -|- m) v' -\-{v" -\- m") v" -\- etc. = fP -\'<lQ~\- ^-^ H~ -^'^ H~ ^tc.
If, accordingly, by the substitutions
p = ax, q = i^x, r = yx, s = dx, etc.
we suppose the functions v, v', v", to become respectively,
— m-\-lx, — m -\- I'x, — m" -{- H'x, etc.,
we shall evidently have the identical equation
(^ ^ -(- Vl' -J- I" I" -\- etc.) XX — {Ifn-^ I'm' -|- Vnf etc.) x=^y.x,
that is,
X ;, _^ r).' -I- I'T + etc. = 0, >t 4- ?. m -f- I'm' + l"m" + etc. = 0 :
hence it must follow that X ^ 0, ^' = 0, "k" = 0, etc. and also x = 0. Then it is
evident, that all the functions V, V V", are such that their values are not
changed, even if the quantities p, q, r, s, etc. receive any increments or decre-
ments whatever, proportional to the numbers a, fi, y, d, etc. : but we have already
mentioned before, that cases of this kind, in which evidently the determination
of the unknown quantities would not be possible, even if the true values of the
functions V, V, V", etc., should be given, do not belong to this subject.
Finally, we can easily reduce to the case here considered, all the others in
which the functions V, V, V", etc. are not linear. Letting, for instance, tt, x, {>,
o, etc., denote approximate values of the unknown quantities jk», §-, r, s, etc., (which
we shall easily obtain if at first we only use v of the fi equations V=3f, V'^=M',
V" z= M", etc.), we will introduce in place of the unknown quantities the others,
p', q, r, /, etc., putting p=zn -\-p>', q-= x -[- 4i r = () -f- /, s = a -|- 5', etc. : the
values of these new unknown quantities will evidently be so small that their
Sect. 3.] any number of observations. 263
squares and products may be neglected, by which means the equations become
Hnear. If, after the calculation is completed, the values of the unknown quanti-
ties p, q, r', s', etc., prove, contrary to expectation, to be so great, as to make it
appear unsafe to neglect the squares and products, a repetition of the same pro-
cess (the corrected values of p, q, r, s, etc. being taken instead of n, /_, q, o, etc.),
will furnish an easy remedy.
181.
When we have only one unknown quantity p, for the determination of which
the values of the functions ap -\- n, ap -\- n, d'p -\- n\ etc. have been found, re-
spectively, equal to M, M', M", etc., and that, also, by means of observations
equally exact, the most probable value of p will be
. am -j- a'm' -|- aW -|- etc.
aa-\- a! a' -j- aV -\- etc. '
putting m, m', m", respectively, for M — n, M' — n', M" — n", etc.
In order to estimate the degree of accuracy to be attributed to this value, let
us suppose that the probability of an error z/ in the observations is expressed by
\Jn
Hence the probability that the true value of p is equal to A -\-p will be propor-
tional to the function
g-AA ({ap—mf-\-{a'p~m'f+{a"p-m"f+ etc.)
if A -\-p is substituted for p. The exponent of this function can be reduced to
the form,
— hh {aa -\- dd -f d'd' -f etc.) ^pp —.2pA-}- B),
in which B is independent of p : therefore the function itself will be propor-
tional to
- — hh(aa-\-a'a'-{-a"a"-\-etD.) p'p'
It is evident, accordingly, that the same degree of accuracy is to be assigned to
the value A as if it had been found by a direct observation, the accuracy of which
would be to the accuracy of the original observations a.s h^{aa-\-dd'^d'd'-\- etc.)
to h, or as ^ (aa -f- dd -j- dW -{- etc.) to unity.
264 DETERMNATION OF AN OllBIT FEOM [BoOK II.
182.
It will be necessary to preface the discussion concerning the degree of accu-
racy to be assigned to the values of the unknown quantities, when there are sev-
eral, with a more careful consideration of the function vv -{- vV -{- v"v" -{- etc.,
which we will denote by W.
I. Let us put
also
a
and it is evident that we have p' = P, and, since
&W__AW '2p'Ap' ^
Ap djo a d/7 '
that the function W is independent of jo. The coefficient a = aa -\- a a -\- a" a" -\-
etc. will evidently always be a positive quantity.
n. In the same manner we will put
also
and we shall have
* ^-^ / = ^' + ^^V + 7'r + d's + etc.,
W'—^^=W",
^'=i^_ii^=:^_^-/, and'-^'=0,
whence it is evident that the function W" is independent both of 7; and q.
This would not be so if ^' could become equal to zero. But it is evident
that W is derived from vv -\- v'v -\-v"v" -\- etc., the quantity p being eliminated
from V, v, v", etc., by means of the equation j»' = 0 ; hence, {^ will be the sum of
the coefficients oi qq in vv, v'v', v"v", etc., after the elimination; each of these
coefficients, in feet, is a square, nor can all vanish at once, except in the case
excluded above, in which the unknown quantities remain indeterminate. Thus
it is evident that §' must be a positive quantity. ♦-
Sect. 3.] any number of observations. 265
III. By putting again,
^^ = / = r + /V + rs + etc., and W' — y= W'%
we shall have
also W" independent of p, and g, as well as r. Finally, that the coeflficient of /"
must be positive is proved in the same manner as in II. In fact, it is readily per-
ceived, that y" is the sum of the coefficients oi rr mvv, vv', v"v", etc., after the
quantities p and q have been eliminated from v, v', v", etc., by means of the equa-
tions p' = 0, / = 0.
IV. In the same way, by putting
ds
we shall have
J^' = / = r' + r'5 + etc., W^=W"'—%,
= S--p-j,q-^r,
W'-'^ independent of p, q, r, s, and d'" a positive quantity.
V. In this manner, if besides p, q, r, s, there are still other unknown quanti-
ties, we can proceed further, so that at length we may have
W= — p'p -\- -XT 44 "h —r ^'^'' -|- Y" ^'^' ~^ ^^^' ~i~ Constant,
in which all the coefficients will be positive quantities.
VI. Now the probability of any system of determinate values for the quan-
tities p, q, r, s, etc. is proportional to the function e~*^^; wherefore, the value of
the quantity p remaining indeterminate, the probability of a system of determi-
nate values for the rest, will be proportional to the integral
/e-^''^djo
extended from jt?=:r — oo to jk»=-|-°° ? which, by the theorem of Laplace, becomes
therefore, this probability will be proportional to the function e~'''''-^\ In the
same manner, if, in addition, q is treated as indeterminate, the probability of a
34
260 DETERMINATION OF AN ORBIT FROM [BoOK 11.
system of determinate values for r, s, etc. will be proportional to the integral
extended from q = — co upto§'=-|-oo, which is
or proportional to the function e"*'''*^". Precisely in the same waj^, if r also is
considered as indeterminate, the probability of the determinate values for the rest,
s, etc. will be proportional to the function e~''''^^"', and so on. Let us suppose the
number of the unknown quantities to amount to four, for the same conclusion
will hold good, whether it is greater or less. The most probable value of s will
be — yr,, and the probability that this will differ from the truth by the quantity
a, will be proportional to the function e~^'''^''"'; whence we conclude that the
measure of the relative precision to be attributed to that determination is ex-
pressed by ^d'", provided the measure of precision to be assigned to the original
observations is put equal to unity.
183.
By the method of the preceding article the measure of precision is conven-
iently expressed for that unknown quantity only, to which the last place has
been assigned in the work of elimination ; in order to avoid which disadvantage,
it will be desirable to express the coefl&cient d'" in another manner. From the
equations
p=/
it follows, that p', q', r, s', can be thus expressed by means of P, Q, R, 8,
p' = P
SeCJ, 3.] ANY NUMBER OF OBSERVATIONS. 267
so that % 5(', 23', W, 23", ^" may be determinate quantities. We shall have,
therefore (by restricting the number of unknown quantities to four),
r' r p I 33"^ , s" pile.
Hence we deduce the following conclusion. The most probable values of the
unknown quantities p, q, r, s, etc., to be derived by elimination from the equations
P=z{), Q = Q, R=0, S=0, etc.,
will, if P, Q, R, S, etc., are regarded for the time as indeterminate, be expressed
in a linear form by the same process of elimination by means of P, Q, R, S, etc.,
so that we may have
p = L-\-AP^BQ^CR-^DS-\- etc.
q=.L'-^AP-^B'Q^C'R^D'-S-^ etc.
r=L"^A'P^B"Q^G"R^D"S^ etc.
s=L"'-]-^"P-^B"'Q^-0"'R^D"'S-\- etc.
etc.
This being done, the most probable values of p, q, r, s, etc., will evidently be
L, U, L", 11" , etc., respectively, and the measure of precision to be assigned to
these determinations respectively will be expressed by
J_ J_ J_ _2_
V3' y/^' v/<^'" V/^"" '
the precision of the original observations being put equal to unity. That which
we have before demonstrated concerning the determination of the unknown
quantity s (for which -ryj-, answers to D'"') can be applied to all the others by the
simple interchange of the unknown quantities.
184.
In order to illustrate the preceding investigations by an example, let us sup-
pose that, by means of observations in which equal accuracy may be assumed,
we have found
268 DETERmNATION OF AN ORBIT FROM [BoOK II.
3jo + 2^ — 5r=5
4|.+ ^+4r=21,
but from a fourth observation, to which is to be assigned one half the same
accuracy only, there results
— 2/>+6^ + 6r = 28.
"We will substitute in place of the last equation the following : —
— JO -|- 3 ^ + 3 r = 14,
and we will suppose this to have resulted from an observation possessing equal
accuracy with the former. Hence we have
P = 27jo+ 6^ — 88
^= 6j(?+15^ + r — 70
and hence by elimination,
19899j(? = 49154 + 809P — 324 Q^^R
131 q= 2617— 12 P+ 54§ — P
6633 r = 12707 4- 2P— 9 ^ + 123 i?.
The most probable values of the unknown quantities, therefore, will be
j» = 2.470
^ = 3.551
r = 1.916
and the relative precision to be assigned to these determinations, the precision of
the original observations being put equal to unity, will be
„ / 19899 . ^^
for ^ \/ ^ = 3.69
forr \/^ = 7.34.
Sect. 3.] J^y number of observations.
185.
The subject we have just treated might give rise to several elegant analytical
investigations, upon which, however, we will not dwell, that we may not be too
much diverted from our object. For the same reason we must reserve for another
occasion the explanation of the devices by means of which the numerical calcu-
lation can be rendered more expeditious. I will add only a single remark.
When the number of the proposed functions or equations is considerable, the
computation becomes a little more troublesome, on this account chiefly, that the
coefficients, by which the original equations are to be multiplied in order to ob-
tain P, Q, a, jS, etc., often involve inconvenient decimal fractions. If in such
a case it does not seem worth while to perform these multiplications in the most
accurate manner by means of logarithmic tables, it will generally be sufficient
to employ in place of these multipliers others more convenient for calculation,
and differing but little from them. This change can produce sensible errors in
that case only in which the measure of precision in the determination of the
unknown quantities proves to be much less than the precision of the original
observations.
186.
In conclusion, the principle that the sum of the squares of the differenc'CS
between the observed and computed quantities must be a minimum may, in the
following manner, be considered independently of the calculus of probabilities.
When the number of unknown quantities is equal to the number of the ob-
served quantities depending on them, the former may be so determined as exactly
to satisfy the latter. But when the number of the former is less than that of the
latter, an absolutely exact agreement cannot be obtained, unless the observations
possess absolute accuracy. In this case care must be taken to estabhsh the best
possible agreement, or to diminish as far as practicable the differences. This idea,
however, from its nature, involves something vague. For, although a system of
values for the unknown quantities which makes all the differences respectively
270 DETERMINATION OF AN ORBIT FROM [BOUK II.
less than another sj^stem, is without doubt to be preferred to the latter, still the
choice between two systems, one of which presents a better agreement in some
observations, the other in others, is left in a measure to our judgment, and innu-
merable different principles can be proposed by which the former condition is
satisfied. Denoting the differences between observation and calculation by J,
J', J", etc., the first condition will be satisfied not only if //z/ -j- /I' J' -{- J" J" -\-
etc, is a minimum (which is our principle), but also if J^-\-J'*-\- J"* -\- etc., or
J^ -\- J"^ -\- J"^ -\- etc., or in general, if the sum of any of the powers with an
even exponent becomes a minimuiu. But of all these principles ours is the most sim-
ple ; by the others we should be led into the most complicated calculations.
Our principle, which we have made use of since the year 1795, has lately
been published by Legendke in the work Noiivelles methodes pour la determination des
orbit es des cometes, Paris, 1806, where several other properties of this principle have
been explained, which, for the sake of brevity, we here omit.
If we were to adopt a power with an infinite even exponent, we should be
led to that system in which the greatest dijEferences become less than in any other
system.
Laplace made use of another principle for the solution of linear equations the
number of which is greater than the number of the unknown quantities, which
had been previou.sly proposed by Boscovich, namely, that the sum of the errors
themselves taken positively, be made a minimum. It can be easily shown, that a
system of values of unknown quantities, derived from this principle alone, must
necessarily* exactly satisfy as many equations out of the number proposed, as
there are unknown quantities, so that the remaining equations come under consid-
eration only so far as the}- help to determine the choice : if, therefore, the equation
V ^=. M, for example, is of the number of those wdiich are not satisfied, the sys-
tem of values found according to this principle would in no respect be changed,
even if any other value N had been observed instead of 31, provided that, denot-
ing the computed value by n, the difierences M — n, N — n, were affected by the
same signs. Besides, Laplace qualifies in some measure this principle by adding
* Except the special cases in which the problem remains, to some extent, indeterminate.
Sect. 3.] any number of observations. 271
a new condition : he requires, namely, that the sum of the differences, the signs
remaining unchanged, be equal to zero. Hence it follows, that the number of
equations exactly represented may be less by unity than the number of unknown
quantities ; but what we have before said will still hold good if there are only
two unknown quantities.
187.
From these general discussions we return to our special subject for the sake
of which they were undertaken. Before the most accurate determination of
the orbit from more observations than are absolutely requisite can be com-
menced, there should be an approximate determination which will nearly satisfy
all the given observations. The corrections to be applied to these approximate
elements, in order to obtain the most exact agreement, will be regarded as the
objects of the problem. And when it can be assumed that these are so small
that their squares and products may be neglected, the corresponding changes,
produced in the computed geocentric places of a heavenly body, can be obtained
by means of the differential formulas given in the Second Section of the First
Book. The computed places, therefore, which we obtain from the corrected ele-
ments, will be expressed by linear functions of the corrections of the elements,
and their comparison with the observed places according to the principles before
explained, will lead to the determination of the most probable values. These
processes are so simple that they require no further illustration, and it appears at
once that any number of observations, however remote from each other, can
be employed. The same method may also be used in the correction of the parcb-
holic orbits of comets, should we have a long series of observations and the best
agreement be required.
188.
The preceding method is adapted principally to those cases in which the
greatest accuracy is desired: but cases very frequently occur where we may,
without hesitation, depart from it a little, provided that by so doing the calcula-
272
DETERMINATION OF AX ORBIT FROM
[Book II.
tion is considerably abridged, especially when the observations do not embrace a
groat interval of time ; here the final determination of the orbit is not yet
proposed. In such cases the following method may be employed with great
a'^viuitage.
Let complete places L and L' be selected from the whole nmnber of observa-
tions, and let the distances of the heavenly body from the earth be computed
from the approximate elements for the corresponding times. Let three hypothe-
ses then be framed with respect to these distances, the computed values being
retained in the first, the first distance being changed in the second hypothesis,
and the second in the third hypothesis ; these changes can be made in proportion
to the uncertainty presumed to remain in the distances. According to these
three hypotheses, which we present in the following table,
Hyp. I.
Hyp. II.
Uyp. ni.
Distance * corresponding to the first place,
Distance corresponding to the second place,
D
D'
D'
D
iX + S
let three sets of elements be computed from the two places X, L', by the methods
explained in the first book, and afterwards from each one of these sets the geo-
centric places of the heavenly body corresponding to the times of all the remain-
ing observations. Let these be (the several longitudes and latitudes, or right
ascensions and declinations, being denoted separately),
in the first set .... ilf. M', M", etc.
in the second set . . . M-\- a, M' -\- a', M"-]- a", etc;
in the third set . . . . M-\- /?, M' -\- (V, M"-\- fi", etc.
Let, moreover, the observed
places be respectively N, N', N", etc.
Now, so far as proportional variations of the individual elements correspond
* ?t will be still more convenient to use, instead of tlie distances themselves, the logarithms of the
curtate distances.
Sect. 3.] any number of observations. 273
to small variations of tlie distances D, D', as well as of the geocentric places
computed from tliem, we can assume, that the geocentric places computed from
the fourth system of elements, based on the distances from the earth D -\-xd,
D' -^y^', are respectively M^ax^^y, M -\-a'x-^ (i'y, M" -\- a"x -)- fy, etc.
Hence, x, y, will be determined, according to the preceding discussions, in such a
manner (the relative accuracy of the observations being taken into account), that
these quantities may as far as possible agree with N, N', N'\ etc., respectively.
The corrected system of elements can be derived either from L, I! and the dis-
tances D -^x^, D' -\-x8', or, according to well-known rules, from the three first
systems of elements by simple interpolation.
189.
This method differs from the preceding in this respect only, that it satisfies
two geocentric places exactly, and then the remaining places as nearly as possi-
ble ; while according to the other method no one observation has the preference
over the rest, but the errors, as far as it can be done, are distributed among all.
The method of the preceding article, therefore, is only not to be preferred to the
former when, allowing some part of the errors to the places L, L', it is possible to
diminish considerably the errors in the remaining places : but yet it is generally
easy, by a suitable choice of the observations L, L', to provide that this difference
cannot become very important. It will be necessary, of course, to take care that
such observations are selected for L, L', as not only possess the greatest accuracy,
but also such that the elements derived from them and the distances are not
too much affected by small variations in the geocentric places. It will not, there-
fore, be judicious to select observations distant from each other by a small inter-
val of time, or those to which correspond nearly opposite or coincident heliocen-
tric places.
35
FOURTH SECTION.
ON THE DETERSIINATION OF ORBITS, TAKING INTO ACCOUNT THE
PERTURBATIONS.
190.
The perturbations which the motions of planets suffer from the influence of .
other planets, are so small and so slow that they only become sensible after a
long interval of time ; within a shorter time, or even within one or several entire
revolutions, according to circumstances, the motion would differ so little from the
motion exactly described, according to the laws of Kepler, in a perfect ellipse,
that observations cannot show the difference. As long as this is true, it would
not be worth while to undertake prematurely the computation of the perturba-
tions, but it will be sufficient to adapt to the observations what we may call an
osculating conic section: but, afterwards, when the planet has been accurately
observed for a longer time, the effect of the perturbations will show itself in such
a manner, that it will no longer be possible to satisfy exactly all the observations
by a purely elliptic motion ; then, accordingly, a complete and permanent agree-
ment cannot be obtained, unless the perturbations are properly connected with
the elll])tic motion.
Since the determination of the elliptic elements with which, in order that the
observations may be exactly represented, the perturbations are to be combined,
supposes a knowledge of the latter ; so, inversely, the theory of the perturlDations
cannot be accurately settled unless the elements are already very nearlj^ kno-\va :
the nature of the case does not admit of this difficult task being accomplished
with complete success at the first trial : but the perturbations and the elements
can be brought to the highest degree of perfection only by alternate corrections
(274)
Sect. 4.] ON the determination of orbits. 275
often repeated. Accordingly, the first theory of perturbations will be constructed
upon those purely elliptical elements which have been approximately adjusted to
the observations ; a new orbit will afterwards be investigated, which, with the
addition of these perturbations, may satisfy, as far as practicable, the observa-
tions. If this orbit differs considerably from the former, a second determination
of the perturbations will be based upon it, and the corrections will be repeated
alternately, until observations, elements, and perturbations agree as nearly as
possible.
191.
Since the development of the theory of perturbations from given elements is
foreign to our purpose, we will only point out here how an approximate orbit
can be so corrected, that, joined with given perturbations, it may satisfy, in
the best manner, the observations. This is accomplished in the most simple
way by a method analogous to those which we have explained in articles 124,
165, 188. The numerical values of the perturbations will be computed from the
equations, for the longitudes in orbit, for the radii vectores, and also for the helio-
centric latitudes, for the times of all the observations which it is proposed to use,
and which can either be three, or four, or more, according to circumstances : for
this calculation the materials will be taken from the approximate elliptic ele-
ments upon which the theory of perturbations has been constructed. Then two
will be selected from all the observations, for which the distances from the earth
will be computed from the same approximate elements : these will constitute the
first hypothesis, the second and third will be formed by changing these distances
a little. After this, in each of the hypotheses, the heliocentric places and the
distances from the sun will be determined from two geocentric places; from those,
after the latitudes have been freed from the perturbations, will be deduced the
longitude of the ascending node, the inclination of the orbit, and the longi-
tudes in orbit. The method of article 110 with some modification is useful in
this calculation, if it is thought worth while to take account of the secular varia-
tion of the longitude of the node and of the inclination. If (i, /5', denote the
heliocentric hi litudes freed from the periodical perturbations; ^, X', the heliocen-
276 ON THE DETERMINATION OF ORBITS, [BoOK II.
trie longitudes; Q, Q-\-J, the longitudes of the ascending node; i,i-{-d, the
inclinations of the orbit ; the equations can be conveniently given in the follow-
ing form : —
tan (3 = tan i sin (X — Q),
— f^] ^. tan (y = tan ^ sin (I' — J — 9,).
tan {i-\-8) ' ^ '
This value of - — t^tt^t acquires all the requisite accuracy by substituting an
approximate value for ^': i and Q> can afterwards be deduced by the common
methods.
Moreover, the sum of the perturbations will be subtracted from the longitudes
in orbit, and also from the two radii vectores, in order to produce purely elliptical
values. But here also the effect, which the secular variations of the place of the
perihelion and of the eccentricity exert upon the longitude in orbit and radius
vector, and which is to be determined by the differential formulas of Section I.
of the First Book, is to be combined directly with the periodical perturbations,
provided the observations are sufficiently distant from each other to make it
appear worth while to take account of it. The remaining elements will be deter-
mined from these longitudes in orbit and corrected radii vectores together with
the corresponding times. Finally, from these elements will be computed the
geocentric places for all the other observations. These being compared with the
observed places, in the manner we have explained in article 188, that set of
distances will be deduced, from which will follow the elements satisfying in the
best possible manner all the remaining observations.
192.
The method explained in the preceding article has been prmcipally adapted
to the determination of the first orbit, including the perturbations : but as soon
as the mean elliptic elements, and the equations of the perturbations have both
become very nearly known, the most accurate determination will be very con-
veniently made with the aid of as many observations as possible by the method
of article 187, which will not require particular explanation in this place. Now
if the number of the best observations is sufficiently great, and a great interval
Sect. 4.] taking into account the perturbations. 277
of time is embraced, this method can also be made to answer in several cases for
the more precise determination of the masses of the disturbing planets, at least
of the larger planets. Indeed, if the mass of any disturbing planet assumed in
the calculation of the perturbations does not seem sufficiently determined, besides
the six unknown quantities depending on the corrections of the elements, yet
another, fj,, will be introduced, putting the ratio of the correct mass to the assumed
one SiS 1 -\- fi to 1 ; it will then be admissible to suppose the perturbations them-
selves to be changed in the same ratio, whence, evidently, in each one of the com-
puted places a new linear term, containing /a, will be produced, the development
of which will be subject to no dif&culty. The comparison of the computed places
with the observed according to the principles above explained, will furnish, at the
same time with the corrections of the elements, also the correction ^. The
masses of several planets even, which exert very considerable perturbations, can
be more exactly determined in this manner. There is no doubt but that the mo-
tions of the new planets, especially Pallas and Juno, which suffer such great per-
turbations from Jupiter, may furnish in this manner after some decades of years,
a most accurate determination of t:he mass of Jupiter ; it may even be possible
perhaps, hereafter, to ascertain, from the perturbations which it exerts upon the
others, the mass of some one of these new planets.
APPENDIX.
1.*
The value of t adopted in the Solar Tables of Hansen and Olufsen, (Copen-
hagen, 1853,) is 365.2563582. Using this and the value of \i,
1
^ 354936'
from the last edition of Laplace's Systhne du Moiide, the computation of Jc is
log2 7r 0.7981798684
Compl. logj{ r.4374022154
Compl. log v/(l + /i) . . . 9.9999993882
\ogJc 8.2355814720
1c = 0.01720210016.
11.
The following method of solving the equation
M=E—ewiE,
is recommended by Encke, Berliner Astronomisches Jahrhich, 1838.
Take any approximate value of E, as e, and compute
M' = £ — /' sin e ,
* The numbering of the Notes of the Appendix designates the articles of the original work to
which they pertain.
(279)
280 APPENDIX.
e" being used to denote e expressed in seconds, then we have
or
M— 3f = E—t — e" (sin E— sin e )
= {E—e) (1 — ecose),
if ^ — e is regarded as a small quantity of the first order, and quantities of
the second order are neglected for the present : — so that the correction of g is
M—3f'
x^ ,
1 — e cos £
and a new approximate value of e is
M— M'
"• 1 — e cos £ '
with which we may proceed in the same manner until the true value is obtained.
It is almost always unnecessary to repeat the calculation of 1 — e cos e. Gener-
ally, if the first s is not too far from the truth, the first computed value of
1 — e cos g may be retained in all the trials.
This process is identical with that of article 11, for X is nothing more than
. d log sin E cos E
dE ~sir^'
if we neglect the modulus of Briggs's system of logarithms, which would subse-
quently disappear of itself, and
_d\o^{e"imE) _ 1
^ d{^'&mE) esin^'
therefore,
^ — l l — eco^E^
and
x = ^{M-\-e'mit — ^) = {31— M') -^. = ^ ^~^^',,
II + 1^ ' ' ^ /<+^ 1 — ecos^'
and the double sign is to be used in such a way that X shall always have the same
sign as cos E. In the first approximations when the value of g differs so much
from E that the differences of the logarithms are uncertaui, the method of this
note will be found most convenient. But when it is desired to insure perfect
agreement to the last decimal place, that of article 11 may be used with
advantage.
APPENDIX.
281
As an illustration, take the data of the example in article 13
Assume e = 326°, and we find
log sine 9.74756 w
log d' 4.70415
log /'sine 4.45171 w
d' sin e = — 28295" = — 7° 51' 35"
Jf' = £ — e" sin e = 333° 51' 35"
Jf— if' = — 4960"
^~^' = — 6226"
1 — e cos 6
And for a second approximation,
£ = 326° — 1° 43' 46" = 324° 16' 14'
log sine 9.7663820?«
log /' 4.7041513
log/' sin £ 4.4705333 w
e"sin£ = — 29548".36=-
M' = 332° 28' 42".36
if— Jf' = 4-12".41
f 15".50
log cos e
9.91857
lege
9.38973
log e cos e
9.30830
1 — 6 cose =.79662
log (1 — ecose)
9.90125
logjf— Jf'
3.69548W
, M—M'
3.79423m
12' 28".36
log(l — ecose)
log(Jf— if')
M—M'
1 — e cos £
which gives
, M—M'
lOff-
° 1 — e cos i
Ez= 324° 16' 14" -f 15".50 = 324° 16'29".50.
9.90356
1.09377
1.19021
Putting
we have
18.
g- = I JO = perihelion distance,
x=:^y/^,
log x = 8.0850664436,
. tan hv-\-k tan^ \v^=.v.'c,
T = — (3 tan ^ y -[- tan^ \ v) j
36
282 APPENDIX.
a table may be computed from this formula, giving v for values of t as the argu-
ment, which will readily furnish the true anomaly corresponding to any time
from the perihelion passage. Table Ha is such a table. It is taken from the
first volume of Annales de V Ohservatoire Imperiale de Paris, (Paris, 1855,) and difiers
from that given in Delambre's Astronomy, (Paris, 1814,) Vol. III., only in the
intervals of the argument, the coefficients for interpolation, and the value of ^
with which it was computed.
The true anomaly corresponding to any value of the argument is found by
the formula
V = vo -{-A^ir— To) -^-A^it — T^f + (t — x^f ^3 -\-Ai{'v — Xq)\
The signs of A-^, A^, A^, are placed before the logarithms of these quantities
in the table.
Burckhabdt's table, Bowditch's Appendix to the third volume of the Mecanique
Celeste, is similar, except that log r is the argument instead of t.
Table Ila contains the true anomaly corresponding to the time from peri-
helion passage in a parabola, the perihelion distance of which is equal to the
earth's mean distance from the sun, and the mass ^ equal to zero. For if we put
^=: 1, jit = 0, we have t^t.
By substituting the value of x in the equation
T = — (3 tan 2 V -\- tan^ h v)
it becomes
T= 27.40389544 (3 tan ^ v + tan^ i v)
= 1.096155816 (75 tan ^y + 25 tan^iyj
and therefore, if we put )c'= 0.912279061,
75 tan ^ y -|- 25 tan^ ^v='/t
log >«' = 9.9601277069
Barker's Table, explained in article 19, contains x' r for the argument v.
The Mean daily motion or the quantity M, therefore, of Barker's Table may be
obtained from table Iln, for any value of v, by multiplying the corresponding
value of T by v! .
The following examples will serve to illustrate the use of the table.
Given, the perihehon distance ^ ^ 0.1 ; the time after perihelion passage
if^ 6'^590997, to find the true anomaly.
APPENDIX. 283
Assuming ^it = 0, we find
7 = 208.42561
To = 200.
T— To= 8.42561
^0 = 110° 24' 46'^69
^(t — To)=+ri4'42".42
A2{'v — rof = — 2'20'U9
A,{<v — rof-=+ 4".76
Ai{T — 'v,f = — 0".16
i^ = IIP 3ri3".52
or
7=208.42561
To =210.
T — To = —1.57439
vo = 111° 50' 16".87
Ai{'r — T;o) = — 12'58".96
^(t — To)2 = — 4".35
As{T — tof = — 0".03
A^{r — r,y = — 0".00
t) = 111° 3ri3".53
The latter form of calculation is to be preferred because the value of t — Tj,
is smaller, and therefore the terms depending on (t — Tq), (t — r^f, (t — Tq)^, are
smaller, and that depending on (t — To)* is insensible ; and it is the onlj form
of which all the appreciable terms are to be found in the table.
Beyond t = 40000, the limit of the table, we can use the formula,
V = 180° — [6.0947259] Q* _ [6.87718] (~) — [7.313] Q*, etc.,
in which the coefl&cients expressed in arc are given by their logarithms.
For T = 40.000, for example, we have
«, = 180° — 10° 6' 6".87 — 3' 8".4 1 — 0".44
= 169°50'44".28.
If V is given, and it is required to find t, we have
284 APPENDIX.
For a first approximation the terms depending on the square and third power
of 1 — To may be neglected, and the value of t — Tq thus found can be corrected
so as to exactly satisfy the equation.
If V exceeds 169°, the formula
T = [1.9149336] tan ^ y -f [1.4378123] tan^ i v
may be used instead of the table.
Thus,fory = 169°50'44".28,
logtan^K. .1.0513610
1.9149336
925.33 2.9662946
logtan^^y. .3.1540830
1.4378123
39074.67 4.5918953
T = 40000.00
This method wiU often be found more convenient than the table, even where
is less than 169°.
35.
Table Va contains Bessel's table here referred to, in a slightly modified
form ; and also a similar table by Posselt, for the coefficients v' and v" in the
formula of article 34,
io = v-\-dv'-\-ddv" -{- d^ v'" + etc.,
it is taken from Encke's edition of Olbers AhJmndlung iiber die leicJdeste wid hequemste
Mdliode die Bahn eims Cometen zu herechnen (Weimar, 1847). The following
explanation of its construction and use is taken from the same work, with
such changes as are needed to adapt it to the notation of the preceding
articles : —
K we put
d- = tan h w
T = tan i V
APPENDIX. 285
the formulas of article 34 become
{i+xr
1 (l+&y
"• ', (1 + &^y "
The second equation, in which v is expressed in terms of 2^, is that given by
Bessel, Monatliche Correspondenz, Yol. XII., p. 197. He also gives the third coeffi-
cient of the series, but has computed a table of only the first two. Posselt, in
the Zeitschrift fur Astronomie und verwandte Wissenschaften, Vol. V., p. 161, has given
the first equation ; he has also given three coefficients of the series, but a table of
the second only, since Bessel's table wUl give the first coefficient simply by
changing the sign. Posselt has changed the sign of the second coefficient also.
Instead of the logarithms as given in the tables of Bessel and Posselt, the
corresponding numbers are given in table Ya, and to avoid large numbers, 0.01
is taken as the unit of (5".
Patting
tan i x = l
the table contains
^ — 10000 (i+a^ ^"^^^^
^ — 10000 (1 + r)* ^Ub^bb
So that when r?; = ec we have
y =: 2^ _|_ ^ ( 100 (5 ) -I- 5 ( 100 (^)2
And when x=^v,
^„=:y_^ (100 (^) — ^'(100(^)2
It seems unnecessary to recompute the table in order to be certain of the
accuracy of the last place, or to extend it further, as its use is limited. For
286 APPENDIX. .
absolute values of d greater than 0.03, and for values of x considerably greater
than 90°, the terms here given would not be sufficient. In such cases the
method of 37 and the following articles should be used.
Example. — For Halley's comet,
logd = 8.5099324, and ^ = 63''.43592, we have
by table II«, to = 99° 36' 55".91
and by table Va, A= i- 417.45 1st cor. + 22' 30".63
5 = + 3.111 2d cor. 4- 32".57
y=99°59'59".ll
which, rigorously, should be 100° ; so that d is in this case too great.
Inversely, we find, for v = 100°,
v = 100° 0'00".00
^=:^ 426.78 1st cor. — 23' 0".83
B = -}- 0.297 2d cor. — rjl
w= 99°36'56".06
which agrees nearly with the preceding value. The change of the table to the
present form has been made under the supervision of D'Arrest.
39.
When table Ha is used instead of Barker's table, w is the value of v, which
corresponds to the argument
at
40.
If we put
^_l-i.A-\-0
the formulas for computing the true anomaly and radius vector are
tan ^v= U„y tan i w
r = E,q sec^ I v.
APPENDIX. 287
Table la for the Ellipse contains log ^^ and log^, for the argument ^, to-
gether with the logarithms of their differences corresponding to a change of a unit
in the seventh decimal place of the argument. It was computed by Prof J. S.
Hubbard, and has been used by him for several years. Since it was in type, a
similar table, computed by Mr. A. Marth, has appeared in the Astronomische Nach-
richten, Vol. XLIII., p. 122. The example of article 43 will furnish an illustra-
tion of its use.
Formulas expressing the differentials of the true anomaly and radius vector
in a very eccentric ellipse, in terms of the differentials of the time of perihelion
passage, the perihelion distance and the eccentricity may be obtained from the
equations of this article.
If we put B =il, C= 0, we have, article 39,
tan htv -\~ ^ tan^ ^ w =^
which, by article 20, gives
dw « 7/ S at J , t J
We also have, article 40,
log tan ^ z; = log tsiu ^ w — Hog (1 — | ^ tan^ i to) -\- log y
and, therefore,
dv cos^^wdw _j_(^7 _|_ f -4 d^
2 sin 1 v COS i w 2 sin i w cos^ ^w (l — ^A) ' 7~ 1 —^A 'J'
dv a cos^ ^ w _j 5 at cos^ ^w ,
%\nv~75i2,n^w{\—^A)"''' 2 ? 75 tan i ,<; (1 _ | ^) ^^
"I" 75 tan i w; (1 — I ^) ^^ I" y "T iZr|7 y
which, by putting
a cos^ i w
K=
75taniw;(l— J^)
T— ^
M-.
N=
2(1 + 9.)
4
(l + e)(l + 9e)
288 APPENDIX.
0- ^^-
p
1-M
10
(l_e)(l + 9e)
is reduced to
ll^^ — KdT—KLtdq-\-\_KMt — ]Sr—OP^de,
observing that dt-=. — dT, if T denotes tlie time of perihelion passage.
If we differentiate the equation
we find
1 -)- e cos 1>
These formulas are given by Nicolai, {Moncdliche Correspondenz, Vol. XXVH,
p. 212). The labor of using them is greatly abridged by the fact that K, L,
M, etc., are computed once for all, and that the quantities needed for this pur-
pose are those required for computing the true anomaly and radius vector.
If the ellipse so nearly approaches the parabola that, in the coefficients, we
may assume
tan hv=.y tan h w
jp- ^ \/ 2 cos" \ V
2 q^i tan \ v
the values oi dv and dr assume a much more simple form. In this case we
should have
Tr- • h^2 cos^ \vsm.\v ^ y/ 2 cos* ^v k\j2q
~ 2^1 tan it; ~ ^t ~~ ^*
r TKT \ n ■n\ • \ 4 20tan*iw"l .
_r 4-1-4 tan^^y ] . _ Stan^v
— L(l-he)(l + 9e)J^''^^ — (l + «)(l + 9e)
and consequently,
'^^"■;^;^/^ + L-?^2(l + 9.)"~(l+«)(l+9e)J^^-
J kU2q
APPENDIX. 289
This form is given by Encke {Berliner Astronomisches Jahr'buch^ 1822, page 184.)
If we put g = 1 in the coefficient of c? e it becomes
^— 20— ^2 5tan^^;.
If we substitute the value of dv in the expression for dr given above, it
may be reduced to the form
dr:= =^= sin ^; t? r-f- cos vdq-{- (/q ■!!!!! 4- yL- r tan^ \v)de.
Si'lq ^ sl2q '
41.
The time t may be found from table Ila, by multiplying the value of r cor-
responding to M' by
^' B
45.
Table la, for the hyperbola is similar to that for the ellipse, and contains
log E^ and log E, for the formulas
tan \v ^=:^E^y tan h. w
r = E^ sec^ hv.
The differential formulas of article 40, of the Appendix, can be applied to
the hyperbola also, by changing the sign of A and of 1 — e in the coefficients.
56.
As the solution here referred to may sometimes be found more convenient
than the one given in articles 53-57, the formulas sufficient for the use of prac-
tical computers are given below.
Using the notation of 60 and the following articles, the expressions for the
rectangular coordinates referred to the equator are, —
a; = r cos u cos Q, — r sin m sin Q> cos i
(1) ^ = r cos M sin 9, cos « -|- ^ sin u cos Q, cos ^ cos e — ^ sin u sin i sin e
5! = r cos M sin Q, sin e -f- ^ sin u cos Q, cos i sin e -|- r sin u sin i cos £
37
290 APPENDIX.
which can be put in the form
x = rsma sin [A -f- u)
(2) ^ = rsmbsm{B -\- u)
g = r sin c sin ( C -f- w)
or
a; = r sin a sin A cos u-{-r sin a cos A sin u
(3) y = r sin b smB cos ^« -|- r sin b cos ^ sin m
g = r sin c sin (7 cos u-{-r sin c cos Csin m
equations (3), compared with (1) give
sin a sm A=: cos Q, sin a cos J. = — sin Q cos e
(4) sin ^ sin 5 ::= sin S2 cos £ sin J cos j5 = cos S2 cos ? cos e — sin ^' sine
sin c sin C' = sin Q, sin e sin c cos C = cos Q cos ^ sin s -\- sin e cos 8 .
By introducing the auxiliary angle E
we shall find
tan^— — ~
cos g^
cotan yl =z — tan g^ cos i
tan gg cos ^ cos £
, /y cos i sin (^4- s)
cotan C = — ^ — ^-7p^^
tan gg cos jfo sin £
•„ „ cosS^ sin Q cos ^
bill I* — : J- }
Sin A cos A
• 7 sin ^ cos £ COS Q, cos i cos £ — sin i sin e
sm 0 — — • "r> — — 75
sm Is COS J}
COS I sin £ -|- sin i cos e
sin C COS O
sin «, sin b, sine are always positive, and the quadrants in which A, B, C are to
he taken, can be decided by means of equations (4).
The following relations between these constants, easdy deducible from the
foregoing, are added, and may be used as checks :
, . sin 5 sin c sin ( C — B)
sin a sin A
APPENDIX. 291
cos a = sin S sin i
cos ^ = — cos Q, sin i cos e — cos / sin e
cos c = — cos Q> sin i sin f -[~ cos / cos £
sin^ a -\- sin^ 5 -|- sin^ <? = 2
cos^ a -|~ cos^ ^ H~ cos^ c ^ 1
cos {A — B)=z — cotan a cotan h
cos [B — 0)z= — cotan h cotan c
cos (J. — ^) = — cotan a cotan c.
58.
K in the formulas of article 56 of the Appendix, the ecliptic is adopted as
the fundamental plane, in which case £ = 0 5 and if we put
n = long, of the perihelion
sha.a=:kx A=zK^ — [n — ^)
mi.l=^ky B = Ky — (tt — Q,)
sinc = >^^ C=^K^ — (tt — Q,)
we shall have
Jc^ sin (^ — (tt — ^ )) = cos g?
^^Cos(^ — (tt — S2)) = — sin g^ cos 2
k^ sin ^ = cos g^ cos (tt — Q,) — sin g^ sin (tt — Q,) cos i
haoB ^ = — [cos g^ sin (tt — Q>)-\- sin g^ cos (jt — 9,) cos 2]
which can easily be reduced to the form,
k^ sin K^ = cos^ i i cos n -\- dv? hi cos {n — 2 Q,)
k^ cos K^=^ — [cos^ I i sin n -\- sin^ h ism {n — 2 $2 )]
and in like manner we should find
ky sin Ky = cos^ i i sin n — sin^ ^ z sin (tt — 2 9,)
kyCosKy=^ cos^ ^2 cosu — sin^ h ^cos(7^ — 2 9)
k^ sin K^ = sin i sin [tt — 9)
k^cos K, = sin i sin (n — 9)
292 APPENDIX.
If these values are substituted in the general expression for coordinates,
ak cos 9 CQ's, K woL E -\- ah sin K {cos JE — e)
and if we put
a cos (p = b
a cos- iicosn 14- tan'' ^ ^ — ^^ 1 = A
_^cos^ i ^• sinTc fl +tan2 ^ . sirK^r--2g)l ^^
L ' sin TT J
acos^ ^ esin^r [l — tan^ |/'^" ^^~'^^^1 = ^^
b cos^ I / cos 7t 1 — tan^ i i '^°'' ^^ ~ 1 = S
L cos ;r J
a sin ^ sin (tt — S2 ) = ^'^
^ sin i cos (jr — S^ ) = B"
the coordinates will be
x = A (cosE — g) + 5 sinE==^ (1 — esecE)^-^ sinE
y = A[ (cos E — e) + ^' sin E = JL' (1 — e sec E) -j- B' sin E
z = A" (cos E — e) 4- 5'' sin E = vl" (1 — e sec E) -f- 5" sin E.
If the equator is adopted as the fundamental plane instead of the ecliptic,
the same formulas may be used, if 9,, n, and i are referred to the equator by
the method of article 55. Thus, if S^^ denote the right ascension of the node
on the equator, for 9,, n, and i, we must use 9,^, 9,^-\-{n — 9) — -^ , and?
respectively.
This form has been given to the computation of coordinates by Prof Peirce,
and is designed to be used with Zech's Tables of Addition and Subtraction Logarithms.
Example. — The data, of the example of articles 56 and 58, furnish
9 = 158° 30' 50".43, n = 122° 12' 23".55, i= 11° 43' 52".89 when the equator
is adopted as the fundamental plane ; and also log b = 0.4288533.
Whence we find
log cos (tt — 2 a ) 9.9853041 n log sin (tt — 2 S2 ) 9.4079143
log sec Tt 0.2732948 m logcosecTt 0.0725618
logtan^^? 8.0234332 logtan^^e 8.0234332
logc 8.2820321 - logc' 7.5039093
add. log -
log cos n
log cos^ h i
log a
logJL
add. log -
log sin n
log cos^ ^ i
log h
logB
0.0082354
C. sub. log -
9.7267052 ?z
log cos Jt
9.9954404
log cos^ i i
0.4423790
\ogb
0.1727600 w
logB'
0.0013836
C. sub. log -,
9.9274382
log sin n
9.9954404
log cos^ h i
0.4288533
log a
0.3531155 w
log^'
293
9.9916052
9.7267052
9.9954404
0.4288533
0.1426041 w
9.9986120
9.9274382
9.9954404
0.4423790
0.3638696
This method may also be used to compute Jc and K for the general formula
of article 57. Thus: —
add. log -
0.0082354
C. sub. log -
9.9916052
log cos TC
9.7267052 w
log cos TT
9.7267052 ?2
log cos^ k i
9.9954404
log cos^ i i
'
9.9954404
log k^ sin Kj.
9.7303810 ??
log ky cos Ky
9.7137508 w
add. log -
0.0013836
G. sub. log -
9.9986120
log sin n .
9.9274382
log sin TT
9.9274382
log cos^ i i
9.9954404
log cos^ i i .
9.9954404
log k^ cos K^
9.9242622^2
log ky sin ^
9.9214906
log tan K^
9.8061188
log tan Ky
0.2077398 J^
log cos K^
9.9254698 w
log sin ^
9.9294058
log^.=.
= 9.9987924
log ^^ =
: 9.9920848
K= 212'
' 36' 56".l
^,= 121° 4/28
".1
It will not be necessary to extend the example to the final expressions for
x,y,z,9,% illustrations of similar apiDlications of the Addition and Subtraction
Logarithms are given in the directions accompanying Zech's Tables.
294 APPENDIX.
If r, h, and I denote the radius vector, the heliocentric latitude and longitude
of any planet, the rectangular coordinates referred to three axes, — of which
that of X is directed towards the vernal equinox, that of z, parallel to the earth's
axis, and that of y, 90° of right ascension in advance of x, — will be as in case 11.
:r = r cos h cos I
y =.r cos h sin I cos e — r sin 5 sin «
0 = r cos b sin £ sin ^-j- r sin h cos £
and by putting
cos b cos /
sin h sin I cos h
sm U = -.-— : =
tan^_ . ,
sin (
sin d cos d
tan b
they assume the following forms convenient for computation :
X = rcosu
?/ = rsm u cos {6 -\- e)
s = r sin u sin (^ -|- e) .
74.
The following are the solutions and examples from the Monatliche Correspon-
denz referred to in this article, adopting the notation of article 74, and using L
to denote the longitude of the Sun.
Given, Q,, L',l,b, i, B, to find u,r, J, and the auxiliary angles A, B, 0, etc.
L
1. <^_^l(^-^l^ ^ tan A sinJ_tan (i^-^ ^ ^^^ ^^
sm (// — l) sin {A -\- t)
cos (L' — Q) sm (i? + b) cos i
o sin(L' — g2)tan5_ ^ sin C sin (Z' — g? ) _x
4. cos(Z--S^)tan5^^^^^ sjn^an (L'-^ ) cos (Z^ - Z) ^ ^^^ ^^
cos (Z — Z) tan ^ sin (Z) -}- Z — /) cos i
APPENDIX. 295
The angle to is to be taken between 0° and 180° when b is positive, and be-
tween 180° and 360° when b is negative. When b = 0°, the body is in one of the
nodes of its orbit, in the ascending node when sin (X' — I) and sin [l — Q.) have
the same sign ; and in the descending node when they have opposite signs.
It is immaterial in which of the two quadrants that give the same tangent,
the auxiliary angles A, B , C, etc., are taken. In the following examples they
are always taken between -\- 90° and — 90".
II.
5. -^^, — T— = tan U
sin ^ sin
{L'-Q)_ r
sin {I — Q ) sin (i — E) sin u R
a J. • • ti r^\ 4. rr cos^sin(X' — g2)sin6 r
6. tan z sm (I — Q) = tSinF . ,„ \. . °°^ — ^ = -=
"^ ^ sm i^M — 0) sin M cos ^ Ji
^ , j_ • n cos (t sin (L' — T) r
7. cos I tan u = tan 6r -;— 77 — ^ ^. — = ^
sin (f — Q, — Cr) cos M Ji
Q tan (I — Q,) , TT • sin ^sin (L' — I) r
O. : = tan jtZ —. — ;rYr ^i — ^ — /? /=r\ ^= "n
cos » Sin (JI — u) sm{l — g^ ) Ji
9. . . ^7/ ^,tan/ sjnlcosil^r
sln^cos(< — hi) sm{u — J) Ji
T n • • / 7 ^ \ i J T?- COS ^sin b cos (L' — Q ) r
10. sm 2 cos (^ — Q) tan u = tan K r-r^ — 7^ ^^ = s
^ ' sin (A — 0) cos u R
-| -| sin C sin (Z — I) , j sin Z r
cos {G-\-L' — l) tan {L! — Q,) cos%~^^'^ ^ sin {u—L) cos (Z'— ^ ) ~~ ^S
1 0 sini)cos(Z^ — a) _ ^ sin^¥ r
cos {p-\-L'— ^ )cos i ~ "^^ ^" sin {u — M) cos (Z' _ ^) — i?
III.
sm
-I ^ R sin Z^sin (Z^ — g^ ) sin z R cos E sin (Z' — g^ ) sin i .
sin (i — Z^)sin6 sin {i — E) sin {I — g^ ) cos 5
-I t- ^cosi^sin(Z' — g^ ) tan i R sin J' sin (Z' — g^ ) sin (/ — g^ ) .
sin {F— b) ~ sin {F—b) — ^
Other expressions for J may be obtained by combining 13 with all the
formulas II.
Examples : —
Given, 9>= 80°59'12".07, r=:281°r34".99, ^=53°23'2".46, i= 10°3r9".55,
/; = — 3° 6' 33".561, log i^= 9.9926158.
296
APPENDIX.
log tan h
log cos [L' — Q.)
C\ogsm{L' — b)
log tan A
^ = — 3° 57'
A-^i--
8.7349698 «
9.9728762 n
0.1313827 n
8.8392287 ?e
2'M36
6° 40' 7''.414
log sin A
log tan [L' — ^
dog sin {A-\-i)
loo* tan u
8.8381955«
9.5620014
0.9350608
9.3352577 ?i
^(^_12°12'37".942
log sin [L' — /)
log tan i
67. log cos (i/ —
log tan B
9.8686173 «
9.2729872
0.0271238^2
9.1687283
^ = 8°23'2r.888
^-f ^ = 5°16'48".327
log cos B
log sin h
logtan(X' — S2)
<7. log sin (5 -I- 5)
C. log cos i
log tan i
9.9953277
8.7343300 »
9.5620014
1.0360961
0.0075025
9.3352577 w
I)
log sin(Z' —
log tan h
a. log sin [L' -
C. log tan i
log tan C
c= — r
0-\-L' — 9,== 192
9.5348776 «
8.7349698 w
0.1313827W
0.7270128
9.1282429 «
39' 7".058
23' 15".864
log sin C
log sin {L' — Q,)
C.logsin(C+i'-
C. log cos i
loff tan u
9.1243583 n
9.5348776 n
) 0.6685194/2
0.0075025
9.3352578 w
log cos (Z' — a )
log tan b
67. log cos (X'—/)
C. log tan i
log tan D
9.9728762 «
8.7349698;?
0.1714973/2
0.7270128
9.6063561 w
i? = — 2r59'51".182
log sin D
log tan [L' — 9,)
log cos (X' — I)
C.logsin(i> + X'-
C. log cos i
log; tan u
9.5735295 n
9.5620014
9.8285027 «
■/) 0.3637217 w
0.0075025
9.3352578 «
D^L' — l^ 205°38'41".348
APPENDIX.
5°.
297
log tan h
log sin (/-
log tan E
8.7349698 w
9.6658973 w
9.0690725
6°4ri2".412
3° 55' 57".138
log sin E
log sin (Z'— £2)
(7. log sin (e — E)
C. log sin u
0gr=::l0gi? + l0g^:
9.0661081
9.5348776 w
1.1637907
0.6746802 «
0.4394566
0.4320724
log tan i
log sin (/ —
log tan F
F—h =
0
9.2729872
9.6658973 ;z
8.9388845 w
•4°5r53".955
r5r20".394
log cos F
log sin h
log sin (Z' — U)
a\ogmi{F—h)
C. log sin u
C. log cos i
9.9983674
8.7343300 w
9.5348776 ?2
1.4896990 7^
0.6746802 w
0.0075025 w
0.4394567
log cos %
log tan u
log tan G
9.9924975
9.3352577 w
9.3277552^
G^ = — 12° 0'27M18
log cos G 9.9903922
log sin (Z' — ^) 9.8686173 72
aiogsin(/— ga— 6^) 0.5705092 7^
C: log cos M 0.0099379
■S2—G^ = — 15° 35' 42^.492
^^g^
0.4394566
8°.
logtan(/— $2) 9.7183744?2 logsin^
9.6717672 7z
log cos ^■ 9.9924975
log sin (Z' — /)
9.8686173 w
log tan ^ 9.7258769 72
C. log sin {H — u)
0.564969572
H=—2^° 0'39".879
C.\og^m{l — 9,)
0.334102772
^— «=— 15°48' r.937
l^gi
0.4394567
38
298
APPENDIX.
log tan h
8.7349698 n
log sin /
9.4991749^
C. log sin i
0.7345153
log sin {L' -
-9)
9.9728762 »
(7. log cos {I—
-9,) 0.0542771
G. log sin [u —
-^)
0.9674054
log tan /
9.5237622 w
1°S5
0.4394565
/=-
- 18° 23' 55".334
u — I =
6° ir 17".392
10°.
log sin i
9.2654847
log cos K
9.9997290
logcos(/ —
-9,) 9.9475229
log sin h
8.7343300 w
log tan u
9.3352577 ?2
log cos [L' -
-9)
9.9728762 w
log tan K
8.5482653 w
0. log sin {K-
^h)
1.7225836
K=
= — 2°r26".344
C. log cos u
0.0099379
K—b^
= Vb' 7".217
i°si
0.4394567
log sin C
log sin [L' — /)
C.logcos(C-f-r— /) 0.1156850 ?e
6'.logtan(i:'— a) 0.4379986
a log cos/ 0.0075025
log tan Z 9.5541617 n
i: = — 19° 42' 32^.533
u — L= 7°29'54".591
12°.
DJ^L'—9,=^M^° 2' 3r'.738
log sin Z> 9.5735295 ?i
logcos(Z' — S2) 9.9728762 n
^.logcos(Z>+X'—S) 0.0002536 «
CAogcoBi 0.0075025
logtanil/(=X) 9.5541618 w
11°.
C + r — /=219°59'25".474
9.1243583 ^^
9.8686173 «
log sin L
0. log sin [u — L)
{7.log cos (i' — 9>
logr
log sin u
log sin i
C'. log sin 5
loo;//
13°
9.5279439 w
0.8843888
0.0271238 »
0.4394565
0.4320724
9.3253198 m
9.2654847
L2656700W
02885469
APPENDIX. 299
76.
If in the equations of article 60,
X — X = // cos d cos a
y — F=//cos^sina
z — Z=. J Bind
a denoting the right ascension, and d the decHnation, we suppose X, Y, Z known,
we have
dx=^ cos a cos d d J — z/ sin a cos d da — J cos a sin d dd
d y = &hia cos d d J -]- J cos cc cos d da — J sina smd dd
d s = smd dJ-\-/l cos d dd.
Multiply the first of these by sin a , and subtract from it the second multiplied by
cos a , and we find
// cos d da = — dxwia -\~ dy sin a .
Multiply the first by cos a and add to it the second multiphed by sin a , and
we find
dx cos a -{-dy sin a = cosd d J — J sin d dd.
Multiply this equation by — sin d and add it to the third of the differential equar
tions above multiplied by cos d and we find
— dx cos « sin (^ — dy sin a sin ^ -|- ds cos d =J dd
and, therefore,
r> , sin a ^ , cos a -,
COS 0 «t a = — dx -\ J- dy
-, rv COS a. sin 5 , sin a sin 5 , , cos 5 ,
d8 = ^ dx ^ — dy^-j-dz.
From the formulas of article 56 of the Appendix are obtained
dx X dy y dz z
dr r' dr ?• ' dr r'
—^■=x cotan (^ + m) , ^ =y cotan {B^u), -£ = 2 cotan ( 6^+ u)
dx . dv . , dz .
-— =: a; sm ^( COS « , -^=:r smw coso, ^-. = r smwcos e,
di ^ di ' a% '
and the partial differentials
dx . dy dz .
jT-== — ycose — ssme, ^ = a; cose, — ^i?;sm6
300 APPENDIX.
whence
dx=^-dr -j-^cotan [A -\-u) c? y -|- a; cotan ( J. -|- ?<) dn
— [x cotan {A -\- u) -\- y 0,0?, B -\- z Bux t] d Q, -\- r miu c(i& a di
dy = ^-dr -\- y cotan (^-f-^) dv -\-y Q.oi-d,n [B -\-ii) dn
— \_y cotan {B -\-u) — x cos i\d9,-\-r^\r\. u cos h di
dz = -dr — z cotan [C -\-u) dv -\- z cotan [C -\- u) d tc
— [z cotan ( C-\- u) — xwa.B]d Q,-\- r sin u cos c di.
These formulas, as well as those of 56 may be found in a small treatise
Ueler die Differeniialformeln fur Comete^i-Bahnen, etc., by G. D. E. Weyer, (Berlin,
1852). They are from Bessel's Abhandhng iiher den Olbers'schen Cometen.
90.
Gauss, in the Berlimr Astronomisches Jakrbuch for 1814, p. 256, has given an-
other method of computing \, and also C of article 100. It is as follows : —
We have
^~"** 6 ' 9X— X
This fraction, by substituting for X. the series of article 90, is readily trans-
formed into
t_ 8 „/. , 2.8 . 3.8.10 . . 4.8.10.12 5.8. 10 . 12. 14 , \
^~105^V^ + ~9"^+ 9.11 ^+ 9.11.13 -^ + 9.11.13.15 ^ "T ^tC. j
Therefore, if we put
we shall have
^-l + ^gi^^ + i|^V + etc.,
xX-\X^^- = -^\^A:^
by means of which ^ can always be found easily and accurately.
APPENDIX. 301
For C, article 100, it is only necessary to write z in place of x in the pre-
ceding formulas.
A may be computed more conveniently by the following formula : —
A_[i. — x) V^i-2.9^-r2.4.9.11^ + 2.4.6.9.11.13^ i" etc. j
142.
Prof. Encke, on the 13th of January, 1848, read a paper before the Eoyal
Academy of Sciences at Berlin, entitled Ueier den Ausnahmefall einer doppelten
Bahnbestimmung aus denselben drei geocentrischen Oertern, in which he entered into a
full discussion of the origin of the ambiguous case here mentioned, and the
manner in which it is to be explained. The following paragraphs, containing
useful instructions to the practical computer, embody the results of his in-
vestigation : —
By putting
5^ = ((y4-a),
Equation IV., 141, becomes, for / > ^
m sin* z = sin [z — q)
and for / <[ iZ'
m sin* z = sin {z -\- q)
m is always positive.
The number and the limits of the roots of this equation may be found by
examining both forms.
Take the first form, and consider the curves, the equations of which are
y -=. m sin* s, y' = sin {z — q)
y and y' being ordinates, and z abscissas.
The first differential coefficients are
-f^im sin^ z cos z, -/- = cos [z — q\
302 appe:ndix.
There will, therefore, be a contact of the curves when we have
m sin* g = sin (^ — q)
and
4 m sin^ s cos z = cos (z — q)
or when
4 sin (s — q) cos s = cos (s — q) sin ^
which may be more simply written
sin (2 s — q) = ^ sin q.
When the value of s deduced from this equation satisfies
m sin* ^ ^ sin (^ — q)
then there is a contact of the curves, or the equation has two equal roots. These
equal roots constitute the limits of possibility of intersection of the curves, or the
limits of the real roots of the equation.
For the delineation of both curves it is only necessary to regard values of
s — q between 0° and 180°, since for values between 180° and 360° the solution
is impossible; and beyond 360° these periods are repeated.
The curve
/ = sin (s — q)
is the simple sine-curve, always on the positive side of ?/', and concave to the axis of
abscissas, and has a maximum for
z — q= 90°.
The cm-ve
^ = sin*0
is of the fourth order, and since it gives
—^ =z 4 «^ sin^ s cos ^ = msm2s — i m sin 4 s
az
3-5^ = 12 m sin^ s cos^ s — 4 m sin* s
= 4 m sin^^ (1 4- 2 cos 2 2) = 2 m (cos 2s — cos 4 2)
-7-4 = — 4 w (sin 2 g — 2 sin 4 s)
dz*
it has a maximum for
- , = — 8 m (cos 2z — 4 cos 4 2)
dz* ^ '
90°
APPENDIX. 303
and a point of contrary flexure for
z = 60°, and z = 120°.
From s = 0° to s = 60°, it is convex to the axis of abscissas, from 60° to
120° it is concave, and convex from 120° to 180°.
For osculation, the three equations,
m sin* 0 = sin [z — q)
4 m sin^ z cosz= cos {z — g)
4:msm^z{l -\-2 cos20)= — sm{z — q)
must coexist, or
m sin^z = sin (z — q)
sin {2z — q) = ^sinq
cos2s=: — |.
In this case we should have
sin (2 s — S') = I cos ^ -j- 1 ^i^ ^y
consequently,
tan q = ^
and
sin^ = |,
or
g = 45° + isin-i|.
From these considerations we infer that for the equation
m sin* z = sin (z — q)
or even when it is in the form
m^ sin® z — 2 m cos q sin^ z -\- sin^ z — sin^ ^ = 0
of the eighth degree, there can only be four real roots ; because, in the whole
period from z — ^=0°to z — q = 360°,only four intersections of the two curves
are possible on the positive side of the axis of ordinates.
Of these, three are between 0=0° and 0^180°, and one between 180"
and 180° -\- q ; or, inversely, one between 0° and 180°, and three between 180°
and lSO°-\-q; consequently, there are three positive and one negative roots, or
three negative and one positive roots for sin z.
304 APPENDIX.
Contact of the curves can exist only when for a given value of q,
/ = iq-\-^ sin~^ I sin q
and
, sin (/ — g)
sin* z
If the contact of the curve of the fourth order with the sine-curve is with-
out the latter, then will m constitute the upper limit, — for 771 greater than this
values of the roots will be impossible. There would then remain only one jjositive
and one negative root.
If the contact is within the sine-curve, then will the corresponding ni" con-
stitute the lower limit, and for m less than this, the roots again would be re-
duced to two, one positive and one negative.
If q is taken negative, or if we adopt the form
m sin* 2 = sin [z -\- q)
180° — z must be substituted for 0.
The equation
rr^ sin^ z — 2 w^ cos q sin^ z -\- sin^ z — sin^ q=^
shows, moreover, according to the rule of Descartes, that, of the four real
roots three can be positive only when q, without regard to sigil is less than
90°, because m is always regarded as positive. For q greater than 90°, there is
always only one real positive root. Now since one real root must always cor-
respond to the orbit of the Earth, that is, to r = JR' ; and since sin d', in the
equation, article 141, —
R' sins'
sm z = 7 —
r
is always positive, so that it can be satisfied by none but positive values
of z; an orbit can correspond to the observations only when three real roots are
positive, or when q without regard to its sign is less than 90°. These limits are
still more narrowly confined, because, also, there can be four real roots only
when in lies between m' and 771", and when we have
f sin ^ < 1, or sin ^ < f , ^ < 36° 52' 11".64
in order that a real value of / may be possible.
APPENDIX. 305
Then the following are the conditions upon which it is possible to find a
planet's orbit different from that of the earth, which shall satisfy three complete
observations.
First. The equation
m sin* g = sin [z -\- q)
must have four real roots. The conditions necessary for this are, that we must
have, without regard to sign,
sin^<|
and m must lie between the limits m' and m".
Second. Of these four real roots three must be positive and one negative.
For this it is necessary that cos q should remain positive for all four of those
values for which
sin^<±|,
the two in the second and third quadrants are excluded, and only values between
— 36° 52' and + 36° 52' are to be retained.
If both these conditions are satisfied, of the three real positive roots, one must
always correspond to the Earth's orbit, and consequently will not satisfy the
problem. And generally there will be no doubt which of the other two will
give a solution of the problem. And since by the meaning of the symbols, arti-
cles 139, 140, we have
sin z sin {8' — z) sin §'
not only must s and d' be always less than 180°, but, also, sin(d' — z) must be
positive, or we must have
d'>z.
If, therefore, we arrange the three real positive roots in the order of their
absolute magnitudes, there may be three distinct cases. Either the smallest root
approaches most nearly the value of d', and corresponds, therefore, to the Earth's
orbit, in which case the problem is impossible ; because the condition d' ^z can
never be fulfilled. Or the middle root coincides with d', then will the problem
be solved only by the smallest root. Or, finally, the greatest of the three roots
differs least from d'. in which case the choice must lie between the two smaller
39
306 APPENDIX.
roots. Each of these will give a planetary orbit, because eacli one fulfils all
the conditions, and it will remain to be determined, from observations other than
the three given ones, which is the true solution.
As the value of m must lie between the two limits m and m", so also must
all four of the roots lie between those roots as limits which correspond to m and
m". In Table TV a. are found, therefore, for the argument q from degree to degree,
the roots corresponding to the limits, arranged according to their magnitude, and
distinguished by the symbols z^, z^, 0™, z". For every value of m which gives a
possible solution, these roots will lie within the quantities given both for m and
m", and we shall be enabled in tbis manner, if d' is found, to discern at the first
glance, whether or not, for a given m and q, the paradoxical case of a double orbit
can occur. It must, to be sure, be considered that, strictly speaking, d' would
only agree exactly with one of the s's, when the corrections of P and Q belong-
ing to the earth's orbit had been employed, and, therefore, a certain difference
even beyond the extremest limit might be allowed, if the intervals of time should
be very great.
The root z", for which sin z is negative, always falls out, and is only intro-
duced here for the sake of completeness. Both parts of this table might have
been blended in one with the proviso of putting in the place of z its supplement ;
for the sake of more rapid inspection, however, the two forms sin [z — q) and
sin (.i -|- q) have been separated, so that q is always regarded as positive in the
table.
To explain the use of Table IVa. two cases are added ; one, the example of Ceres
in this Appendix, and the other, the exceptional case that occurred to Dr. Gould,
in his computation of the orbit of the fifth comet of the year 1847, an account of
which is given in his Astronomical Journal, Vol. I., No. 19.
I. In our example of Ceres, the final equation in the first hypothesis is
[0.9112987] sin* .2 = sin {z — 7° 49' 2".0)
and
d' = 24° 19' 53".34
the factor in brackets being the logarithm. By the table, the numerical factor
lies between ni and m", and this d' answers to z", concerning which there can be
no hesitation, since s"" must lie between 10° 27' and 87° 34'. Accordingly, we
APPENDIX. 307
have only to choose for the s' which occurs in this case, and which, as we per-
ceive, is to be sought between 7° 50' and 10° 27'.
The root is in fact
2^=1" 59' 30".3,
and the remaining roots,
z^= 26 24 3
, 0^ = 148 2 35
0^=187 40 9
are all found within the limits of the table.
2. In the case of the fifth comet of 1847, Dr. Gould derived from his first
hypothesis the equation
[9.7021264] sm'z = sin {s + 32° 53' 28".5).
He had also
(J' = 133° 0' 31".
Then we have sin g <C j, and the inspection of the table shows that the factor
in the parenthesis lies between n/ and m" ; therefore, there will be four real roots,
of which three will be positive. The given d' approximates here most nearly to
2!™, about which, at any rate, there can be no doubt.
Consequently, the paradoxical case of the determination of a double orbit
occurs here, and the two possible values of s will lie between
88° 29' — 105° 59'
and
105 59 —131 7
In fact, the four roots are,
s>= 95°
31' 43".5
g" = 117
31 13 .1
0- = 137
38 16 .7
0- = 329
58 35 .5.
By a small decrease of m without changing q, or by a small decrease of q
without changing m, a point of osculation will be obtained corresponding to
nearly a mean between the second and third roots; and on the contrary, by a
small increase of m without changing q, or a small increase of q without changing
m, a point of osculation is obtained corresponding to nearly a mean between the
first and second roots.
308 APPENDIX.
We have, therefore, the choice between the two orbits. The root used bj Dr.
Gould was 2°, which gave him an ellipse of very short period. The other obser-
vations showed him that this was not the real orbit. M. D'Arrest was involved in
a similar difficulty with the same comet, and arrived also at an ellipse. An ellipse
of eighty-one years resulted from the use of the other root.
" Finally, both forms of the table show that the exceptional case can never
occur when d' < 63° 26'.
" It will also seldom occur when d' < 90°. For then it can only take place
with the first form sin (g — q), and since here for all values of q either the limits
are very narrow, or one of the limits approximates very nearly to 90°, so it will
be perceived that the case where there are two possible roots for d' < 90° will
very seldom happen. For the smaller planets, therefore, which for the most part
are discovered near opposition, there is rarely occasion to look at the table. For
the comets we shall have more frequently d' > 90° ; still, even here, on account
of the proximity to the sun, d' > 150° can, for the most part, be excluded. Con-
sequently, it will be necessary, in order that the exceptional case should occur,
that we should have in general, the combination of the conditions d' > 90° and
q between 0° and 32° in the form sin (2 — q), or between 22° and 36° 52' in the
form sin (s -j- q)."
Professor Peirce has communicated to the American Academy several methods
of exhibiting the geometrical construction of this celebrated equation, and of
others which, like this, involve two parameters, some of which are novel and
curious. In order to explain them, let us resume the fundamental equation,
m sin^ s = sin {s — q).
1. The first method of representation is by logarithmic curves ; the logarithm
of the given equation is
log m -|- 4 log sin s = log sin {s — q).
If we construct the curve
^ =z 4 log sin 0,
APPENDIX. 309
and also the same curve on another scale, in which y is reduced to one fourth of
its value, so that
y := log sin s,
it is plain that if the second curve is removed parallel to itself by a distance equal
to q in the direction of the axis of z, and by a distance equal to — log m in the
direction of the axis of y, the value of z on the first curve where the two curves
intersect each other will be a root of the given equation ; for, since the point of
intersection is on the first curve^ its coordinates satisfy the equation,
^ =: 4 log sin z,
and because it is on the second curve its coordinates satisfy the equation,
y -[- log m = log sin (s — q)\
and by eliminating y from these two equations we return to the original equation,
m sin*0 = sin {z — q).
A diagram constructed on this principle is illustrated by figure 5, and it will
be readily seen how, by moving one curve upon the other, according to the
changeable values of q and ?n, the points of intersection will be exhibited, and also
the limits at which they become points of osculation.
On this and all the succeeding diagrams, we may remark, once for all, that
two cases are shown, one of which is the preceding example of the planet Ceres,
in which the four roots of the equation will correspond in all the figures to the
four points of intersection D, D', 17', D'", and the other of which is the very
remarkable case that occurred to Dr. Gould, approaching the two limits of
the osculation of the second order, the details of which are given in No. 19 of his
Astronomical Journal, and the points of which are marked on all our diagrams
G-, Cr , Cr , br .
2. The second method of representation is by a fixed curve and straight line,
as follows.
[a.) The fundamental equation, developed in its second member, and divided
by m Go&z, assumes the form
sin* z cos <7 / , , V
= — - ( tan z — tan a)
cos 2 m ^ ^'
By putting
q, a
X = tan z,b=z tan r " — ^ ^^
310 APPENDIX.
the roots of the equation will correspond to the points of intersection of the
with the straight line
2/ = a (x — b). [Figs. 6 and 6'.]
It will be perceived that the curve line, in this as in all the following cases
under this form, is not affected hj any change iii the values of m and q, and that
the position of the straight line is determined by its cutting the axis of x at
the distance tan q from the origin, and the axis of ^ at the distance — ^^
from the origin. The tangent of its inclination to the axis is obviously equal to
^^, which may in some cases answer more conveniently for determining its
position than its intersection with the axis of ^.
(b.) The development of the fundamental equation divided by m sin g, is
sin^ s = -^ ( CO tan q — cotan s) ;
and by putting
X = cotan 2
b = cotan g-
m
the roots of the equation correspond to the intersection of the curve
y = sin^ s = (1 -\- x^)~^
with the straight line
t/ = a{b — x). [Fig. 7.]
The position of the straight line is determined by its cutting the axis of x at a
distance equal to cotan q from the origin, and the axis of y at a distance equal to — ^
from the origin. This form of construction is identical with that given by M.
Binet in the Journal de FEcole Poly technique^ 20 Cahier, Tome XIII. p. 285. His
method of fixing the position of the straight line is not strictly accurate. This
mode of representation is not surpassed by either of the others under this form.
[c.) The fourth root of the fundamental equation developed, and divided by
cos [z — q\ assumes the form
i^mcos^(tan(« — q) ^ tan^?) = ^['os{z-q)^-
APPENDIX. 311
By putting
X = tan {z — q)
b = tan q
a =^ \l m cos q
the roots of the equation correspond to the intersection of the curve
_ ^ (sin {z — q)) ^ ^^ Q I ^^1
^ cos (z — q) \ \^ )
with the straight line
y = a{x^h). [Fig. 8.]
The straight line cuts the axis of ;p at a distance equal to — tan q, and the axis
of ^ at a distance equal to y' m sin q, from the origin.
[d.) The development of the fourth root of the fundamental equation divided
by sin {z — q) is,
3
'^ m mxq (cotan [z — q) -\- cotan q) = cosec (s — q).
By putting
^ = cotan {z — q)
h = cotan q
a = ^ m sin q
the roots of the equation correspond to the intersection of the curve
with the straight line
^ = a{x-\-b). [Figs. 9 and 9'.]
The straight line cuts the axis of a; at a distance equal to — cotan q, and the
axis of ?/ at a distance equal to i^ m cos q, from the origin.
{e.) From the reciprocal of the fundamental equation multiplied by m, its
roots may be seen to correspond to the intersection of the curve
r = cosec^ z
with the straight line
r = m cosec [z — q). [Figs. 10 and 10'.]
Both these equations are referred to polar coordinates, of which r is the radius
vector, z the angle which the radius vector makes with the polar axis, m the dis-
tance of the straight line from the origin, and q the inclination of the line to the
polar axis.
312 APPENDIX.
(/). From tlie reciprocal of the fourth root of the fundamental equation, its
roots may be seen to correspond to the intersection of the curve
with the straight line
which
r = cosec* (jp
= ^-cosec(y4-^),
--z-q. [Fig. 11.]
Both these equations are referred to polar coordinates, of which y is the
angle which the radius vector r makes with the polar axis, y/ - the distance of the
straight line from the origin, and q the inclination of the line to the polar axis.
3. The tliird method of representation is by a curve and a circle.
(«.) The roots of the fundamental equation correspond to the intersection
of the curve
with the circle
^=^sin(^-4 [Fig. 12.].
Both these equations are referred to polar coordinates, of which r is the radius
vector, z the ano-le which the radius vector makes with the polar axis, — the
radius of the circle which passes through the origin, and 90° -\- q \% the angle
which the diameter drawn to the origin makes with the polar axis.
(^.) From the fourth root of the fundamental equation it appears that its
roots correspond to the intersection of the equation
r = y' sin g)
with the circle
r=^ m sin (9 -f q) [Fig. 13],
in which (p^{z — q) is the inclination of the radius vector to the polar axis,
^m is the diameter of the circle which passes through the origin, and 90° — q
is the inclination of the diameter drawn through the origin of the polar axis.
In these last two delineations the curve IK I' K' I" incloses a space, within
which the centre of the circle must be contained, in order that there should be
four real roots, and therefore that there should be a possible orbit. The curve
APPENDIX.
313
itself corresponds to tlie limiting points of osculation denoted by Professor Encke's
m' and vd', and the points K and K' correspond to the extreme points of oscula-
tion of the second order, for which Encke has given the values 5' = =f 36° 52'
and w/ = 4.2976, and m' = 9.9999.
On the delineations, /S is the centre of the circle for our example of Ceres,
and /S' the same for Dr. Gould's exceptional case. A careful examination of the
singular position of the point /S" will illustrate the peculiar difficulties attending
the solution of this rare example.
159.
We add another example, which was prepared with great care to illustrate the
Method of Computing an Orbit from three observations published in pamphlet
form for the use of the American Ephemeris and Nautical Almanac ixi 1852. It
furnishes an illustration of the case of the determination of two orbits from the
same three geocentric places, referred to in article 142.
We take the following observations, made at the Greenwich Observatory,
from the volume for the year 1845, p. 36.
Mean Time, Greenwich.
Apparent Eight Ascension.
Apparent Declination.
1845. July 30, Ti 5 lols
Sept. 6, 11 5 56.8
Oct. 14, 8 19 35.9
33°9 51 15J5
3.S2 22 39.30
328 7 51.45
,S'. 23 31 34.60
27 10 23.13
26 49 57.23
From the Naidical Almanac for the same year, we obtain
Longitude of the Sun
from App. Equinox.
Distance from the
Earth.
Latitude of the
Sun.
Apparent Obliquity
of the Ecliptic.
July 30.
Sept. 6.
Oct. 14.
127 40 11.32
164 9 40.85
201 21 12.49
-14.99
-14.06
-12.16
0.0064168
0.0031096
9.9984688
—0.17
+0.21
-1-0.53
27 28.13
28.41
28.05
The computation is arranged as if the orbit were wholly unknown, on which
account we are not at liberty to free the places of Ceres from parallax, but must
transfer it to the places of the earth.
40
!14 APPENDKL
Reducing the observed places of the planet from the equator to the ecliptic,
ve find
Date.
App. Longitude of Ceres.
App. Latitude of Cores.
July 30.
Sept. 6.
Oct. 14.
332 28 28.02
324 35 58.87
321 4 54.55
S. 13 54 52.47
14 45 30.00
13 5 35.33
And also.
Date.
Longitude of Zenith.
Latitude of Zenith.
July 30.
Sept. 6.
Oct. 14.
11 6
4 49
1 4
K 53 26
56 22
58 4
The method of article 72 gives
Date.
Reduction of Longitude.
Reduction of Distance.
Reduction of Time.
July 30.
Sept. 6.
Oct. 14.
+16.32
- 7.10
—26.95
+0.0001368
1421
0907
—0.070
—0.065
—0.071
The reduction of time is merely added to show that it is wholly insensible.
All the longitudes, both of the planet and of the earth, are to be reduced to
the mean vernal equinox for the beginning of the year 1845, which is taken as
the epoch ; the nutation, therefore, being applied, we are still to subtract the
precession, which for the three observations is 28''.99, 34".20, and 39".41, re-
spectively; so that for the first observation it is necessary to add — 43"'.98, for
the second, — 48".26, and for the third, — 5r'.57.
Finally, the latitudes and longitudes of Ceres are to be freed from the aber-
ration of the fixed stars, by subtracting from the longitudes 18".76, 19".69, and
10".40, respectively, and adding to the latitudes — 2.02, +1.72, and -|-4.02,
numbers which are obtained from the following formulas of Prof Peirce : —
d a = m cos (O — a) sec {i
(J /:? = msin (© — a) sin a ;
where O =: sun's longitude, and m ■= aberration of ©.
APPENDIX.
!15
The longitudes of the sun were corrected for aberration by adding 20".06,
20''.21, and 20".43, respectively, to the numbers given in the Nautical
Almanac.
These reductions having been made, the correct data of the problem are as
follows : —
Times of observation.
For Washington Meridian.
Ceres'slong. a, «', a"
latitudes /:J, {i', f
Earth's long. I, I', l"
loffs. of dist. R, R', R"
July 30. 372903.
330° 27' 25".28
— 13 54 54 .49
307 39 43 .66
0.0064753
Sept. 6. 248435.
324 34 50.92
— 14 45 28.28
344 8 45.49
0.0031709
By the formulas of Arts. 136 and 137, we find
7,7,7 ' ' • .
d,d',d" . . . .
log d, d', d" sines
AD,AD\AD"
A'D,A:'iy,AD",
I rr
log £, e', b" sines,
log sin h «'
log cos ^ e'
329° 25' 34".81
28 12 56 .84
9.6746717
199° 45' 41".00
233 54 11 .72
27 32 45 .72
9.6650753
218° 11' 22".38
24 19 53 .34
9.6149131
204° 8'25".14
233 31 23 .54
142 37 25 .44
9.7832221
9.9764767
9.5057163
And by article 138,
log^sin?; .... 6.2654993 w
logTco^t 9.2956278??
wherefore
t=im° 3' 12".63, log ^ . . . 9.2956280
t-\-7'= 38°14'35".01, logsin(^ + /) 9.7916898
log /S' 8.6990834
log ^ sin (^-f-/) . . 9.0873178 '
Whence log tan ((^' — 0) . . . 9.6117656
d' — G = 22° 14' 47".47 and a = 2° 5' 5''.87.
Oct. 14. 132915.
321 3 52.58
— 13 5 31.31
21 19 53.97
9.9985083
194 69 35 .15
61 6 50.78
9.9422976
203° 56' 46".56
199 30 24 .04
115 4 41 .10
9.956992
316 APPENDIX.
By articles 140-143, we find
A" D' — d" = 172° 24' 32".76 log sin 9.1208995 log cos 9.9961773 n
AD' — 8 =175 55 28.30 8.8516890 9.9989004«
A:'D — d" =172 47 20.94 9.0987168
AD — d'^o =177 30 53.53 8.6370904
Aiy' — d =175 43 49.72 8.8718546
A:i)" — d'-^o=m 15 36.57 8.6794373
log a 0.0095516, a =1.0222370
log 5 0.1389045.
Formula 13, which serves as a check, would give log i = 0.1389059. We
prefer the latter value, because sin {A D — id' -|- a) is less than sin {A U'
-8'J^o).
The interval of- the time (not corrected) between the second and third obser-
vations is 37.884480 days, and between the first and second 37.875532 days.
The logarithms of these numbers are 1.5784613 and 1.5783587 ; the logarithm
of k is 8.2355814 ; whence log ^ = 9.8140427, log r = 9.8139401.
We shall put, therefore, for the first hypothesis
2: = logP = ^" =9.9998974
^ = log ^ = (3 r = 9.6269828
and we find
w = 5° 43' 56".13
w4-a = 7 49 2 .00
log Q c sin M = 0.9112987
It is found, by a few trials, that the equation
^ c sin ca sin* s = sin {z -j- T 49' 2".00)
is satisfied by the value
z = T 59' 30".30,
whence log sin z = 9.1431101, and
r'=^?^'= 0.474939.
APPENDIX. 317
Besides this solution, the equation admits of three others, —
z= 26° 24' 3"
^=148 2 35
g=187 40 9
The third must be rejected, because sin z is negative ; the second, because z is
greater than d' ; the first answers to the approximation to the orbit of the earth,
of which we have spoken in article 142 *
The manner of making these trials is as follows. On looking at the table of
sines we are led to take for a first approximation for one of the values, gr = 8"
nearly, or 8° -}- x. Then we have
log sins 9.14356 + 89 a;
logsm*0 6.57424 + 356 :»
log ^csin to 0.91130
logsin(0 — w — (j) . . . 7.48554-1- 356 iP
g_a,_a = 0°10'52''-f-^\V9^
w-l-(T = 7 49 3
2=7 59 55 -f- 1^2 ^j nearly = 8° -|- a?.
For the second approximation, we make
z—T 59' 30'' -f- x ; and have
log sins 9.1431056 + 150 a;^
log sin* s 6.5724224 + 600 «^,
^csinw . . . . . 0.9112987
log sin (s — o — (t) . 7.4837211 + 600 a;'
g _ w — (T = 0° 10' 28".27 + iV ^' nearly.
w + (7 = 7 49 2. 00
z=n 59 30. 27 + iV^'='J'°59'30".30.
The process is the same for the other roots.
* See article 142 of the Appendix.
318 APPENDIX.
Again, by art. 143 we obtain
C= 185^0' 3r.78
r = 189 25 30 .25
log r= 0.4749722
log /'=: 0.4744748
i (^" _|_ u) = 264° 2V 48''.61
^u'' — u) = 2SS 49 5.19
2/ = 6 57 7 .46
2/" == 6 56 32 .68
The sum 2/-|-2/", which is a check, only differs by 0"'.20 from 2/', and the
equation
p __ r sin 2f" __ ?/'
/' sin 2/ w
is sufficiently satisfied by distributing this 0".2 equally between 2/ and 2/", so
that 2/= 6°59'7".36, and 2/"= 6°56'32".58.
Now, in order that the times may be corrected for aberration, the distances
q, q', q" must be computed by the formulas of Art. 145, and then multipHed into
the time 493^ or 0^005706, as follows: —
logr 0.47497
logsin(AD — Q .... 9.51187
comp. log sin ^ 0.32533
log 9
0.31217
log const
7.76054
log of reduction
8.07271
Eeduction =
: 0.011823
log/,
0.47497
log sin [d — z)
9.44921
comp log sin d',
0.38509
log of reduction
0.30927
Reduction, 0.011744.
The constant of aberration is that of M. Struve.
APPENDIX.
log/' 0.47447
logsm(A''D' — n . . . 9.84253
log sin r 0.05770
log of reduction .... 0.37470
Reductions 0.013653
Observations.
Con-ected Times. Intervals.
Logarithms.
I
July 30. 361080
n.
Sept. 6. 236691 37.876611
1.5783596
m.
Oct. 14. 119260 37.882569
1.5784395
319
Hence the corrected logarithms of the quantities &, &" become 9.8140209,
and 9.8139410.
We are now, according to the precept of Art. 146, to commence the determi-
nation of the elements from the quantities/, /, r", 6, and to continue the calcula-
tion so far as to obtain rj, and again from the quantities /", r, /, ^" so as to
obtain rl'.
logi? 0.0011576
logrf' 0.0011552
logP' .... 9.99^9225
log ^ .... 9.6309476
From the first hypothesis, therefore, there results X = 0.0000251, and
Z=: 0.0029648.
In the second hypothesis, we assign to P and Q the values which we find
in the first hypothesis for P' and Q. We put, therefore,
^ = log P= 9.9999225,
j/ = log^= 9.6309476.
Since the computation is to be performed in precisely the same manner as in
the first hypothesis, it is sufficient to set down here its principal results : —
oj 5°43'56'M0
w + (J 7 49 1 .97
log Qc^vciiii 0.9142633
log /
log-;
r 59' 34" 98
. 0.4749037
. 0.7724177
320
APPENDIX.
. 0.7724952
185° 10' 39" 64
189 25 42 .36
logr 0.4748696
loo-r" .• . 0.4743915
i {u + m)
2/' . .
2/. . .
2/" . .
264° 21' 50"
288 49 5
.13 53 58
. 6 57 15
. 6 56 43
.64
.57
82
58
41
In this case we distribute the difference 0".17 so as to make 2/= 6° 51' 15".49
and 2/"= 6° 56' 43".33.
It would not be worth while to compute anew the reductions of the time on
account of the aberration, for they scarcely differ 1" from those which we de-
rived from the first hypothesis.
Further computations furnish
log 7] = 0.0011582, log?]" = 0.0011558, whence are deduced
log P'= 9.9999225, X= 0.0000000
log ^=9.6309955, Y= 0.0000479.
From which it is apparent how much more exact the second hypothesis is than
the first.
For the sake of completing the example, we will still construct the third
hypothesis, in which we shall adopt the values of P' and Q' derived from the
second hypothesis for the values of P and Q.
Putting, therefore,
a; = log Pz= 9.9999225
^ = \ogQ = 9.6309955
the following are obtained for the most important parts of the computation : —
0) 5°43'56".10
(0 -\-a . . .
log Q c sin 0)
log r .
log —
7 49 1 .97
0.9143111
7° 59' 35".02
0.4749031
0.7724168
. 0.7724943
185° 10' 39".69
t;' 189° 25' 42".45
logr 0.4748690
logr" 0.4743909
^{ti'J^u) .... 264° 21' 50".64
^(i^"_M) .... 288 49 5 .57
2/' 13 53 58 .94
2/ 6 57 15 .65
2/" 6 56 43 .49
APPENDIX.
321
The difference 0".2 between 2/' and 2/ -j- If" is divided as in the first
hypothesis, making 2/= 6° 57' 15".55, and 2/"= 6° 56' 43".39.
All these numbers differ so little from those given by the second hypothesis
that it may safely be concluded that the third hypothesis requires no further cor-
rection ; if the computation should be continued as in the preceding hypotheses,
the result would be X=: 0.0000000, r= 0.0000001, which last value must be
regarded as of no consequence, and not exceeding the unavoidable uncertainty
belonging to the last decimal figure.
We are, therefore, at liberty to proceed to the determination of the elements
from 2/', r, r", ^' according to the methods contained in articles 88-97.
The elements are found to be as follows : —
Epoch of the mean longitude, 1845, .... 278° 47' 13".79
Mean daily motion, .... .... 771".5855
Longitude of the perihelion, 148° 27' 49".70
Angle of eccentricity, 4 33 28 .35
Logarithm of the major semi-axis .... 0.4417481
Longitude of the ascending node, .... 80° 46' 36".94
Inchnation of the orbit, 10 37 7 .98
The computation of the middle place from these elements gives
a'= 324° 34' 5r.05, ^'^ — 14° 45' 28".31
which differ but little from the observed values
«'=324°34'50".92,
14°45'28".28.
41
322 APPENDIX.
FORMULAS FOR COMPUTING THE ORBIT OF A COMET.
Given
Mean times of the observations in days, H , f, t'"
Observed longitudes of the comet, a', a", a'"
Observed latitudes of the comet, (i', ft", ft'"
Longitudes of the sun, A, A', A"
Distances of the sun from the earth, R', R", R"'
Required
The curtate distances from the earth, q', q", q"'
Compute
I.
tan^" j^ f— f m sin (a' — A") — tan (i'
sin (a" — A") f — t^ tan /3'" — msin(a'" — A")
and by means of this, approximately,
q"' = Mq\
n.
R"' cos {A"— A) — Rf=zffcos{G — A)
R!" sin {A"— A) =:.gmi{G — A)
g is the chord of the earth's orbit between the first and third places of the earth.
G the longitude of the first place of the earth as seen from the third place.
III.
M— cos {a'"— a') = hcos^ cos {R— a"')
sin («'" — a') = k cos ^ sin {R — a'")
if tan ft"' — tan ft' =i h sin ^.
h is always positive. If iVis a point, the coordinates of which, referred to the
third place of the earth, are
q' cos a', q' sin a', q tan ft,
then are
hq', H, t,
APPENDIX. 323
the polar coordinates of the third place of the comet, (that is, the distance, longi-
tude and latitude,) referred to the point i\^ as the origin.
IV.
cos C cos ( 6^ — ^) = cos 9 ^sm(p=A
cos (^ COS {a' — A') = cos Y R' sin \^'^= B'
cos r cos («'" — A^") = cos v^'" R'" sin ^>"' = B'"
By means of 9 , ^i', ip'", A, B\ B"\ Olbers's formulas, become : —
F ==(>^9'— ^cos9)2+^2
/2 = (q' sec {^' — R' cos x^J 4- B""
r""" = {M^' sec {i'" — R'" cos ^>"'f -f ^'"^
The computation would be somewhat easier by
V.
cos (i'^^f, g cos 9 — /' R! cos »f '= c'
—Kr-=f gcos^i'—f'R cost/^ =c
in which
u=ihq' — g cos 9
VI.
A value of u is to be found by trial which will satisfy the equation
(/ + ,-+ /,)-| _(/+ r"'-Jcf = ^,
in which
log w/= 0.9862673
If no approximate value for q' or for / or r" is otherwise known, by means
of which an approximate value of u can be found, we may begin with
324 ' APPENDIX.
This trial will be facilitated by Table ITIa, which gives fi corresponding to
"?— (/+/")!'
by means of which is found k, which corresponds rigorously to r, r"', and f" — f: —
in which
log x = 8.5366114.
The process may be as follows : For any value of u compute k, /, /", by V,
and with r', /", compute r], with which fi, is to be taken from Table Ilia, and a value
of k is to be computed which corresponds to the /, r"', f — i! used. And m is to
be changed until the second value of k shall agree exactly with that computed
byV.
Then we have
_/ M + g'cosq)
^ — h
vn.
q' cos {a' — A) — R' =zr cos h' cos [t — A)
(/ sin {a' — A) = / cos h' sin [t — A)
q' tan {V = r sin h'
^"' cos {a'" — A") — R" = r'" cos V" cos {t" — A")
q'" sin {a" — A") = r" cos h'" sin {t" — A'')
q"' tan (i'" = r" sin V".
FIRST CONTROL.
The values of /, /", obtained from these formulas, must agree exactly with
those before computed.
t, y ; I" , h'", are heliocentric longitudes and latitudes of the comet.
The motion is direct when f — I' is positive, and retrograde when f' — t is
negative.
APPENDIX. 325
vni.
± tan V = tan i sin {J — ^ )
, tanJ'" — tan&'cos(r'— Z') , . ,,, ^,
=*= sm(r-0 ^ = tanecos^/— S2)
i the inclinatioi] is always positive, and less than 90°. The upper signs are to be
used when the motion is direct j the lower when it is retrograde.
IX.
tan (Z' — S2 ) , IT, „ X tan ill" — g^) . /t-w r^\
— ^ — -. '- = tan (Jj — Q), — ^ r^^= tan (L — Q, ).
cos ^ \ /' cos I ^ '
I! and L'" are the longitudes in orbit.
SECOND CONTROL.
The value of It before computed must be exactly
^ = V^ [r'2 + r""" — 2/ r'" cos {U" — Z' )].
X.
1 cos 1 {L' 7t)
cos {II" — L') cosec \ {L'" — L') sin i {L' — n)
\J r' \j r'" \j q
n, the longitude of the perihelion, is counted from a point in the orbit from which
the distance, in the direction" of the order of the signs, to the ascending node, is
equal to the longitude of the ascending node.
XI.
The true anomalies are
v'=r—7i, v"'=L'"—ii.
With these the corresponding M' and M"' are to be taken from Barker's
Table, and we have then the time of perihelion passage
326 APPENDIX.
in which M' and M'" have the sign of v and v" ; the constant log n is
log n = 0.0398723.
The upper signs serve for direct, the lower for retrograde motion.
For the use of Table TLa instead of Bakker's Table, see Article 18 of the
Appendix.
THIRD CONTROL.
The two values of T, from H, and f, must agree exactly.
XII.
With T, q, 71, Q,, i, I", A', R", compute a" and (i", and compare them with the
observed values. And also compute with these values the formula
tan S"
sin {a — A )
If this value agrees with that of m of formulas I., the orbit is exactly deter-
mined according to the principles of Olbers's Method. That is, while it satisfies
exactly the two extreme places of the comet, it agrees with the observations in
the great circle which connects the middle place of the Comet with the middle
place of the Sun.
If a difference is found, M can be changed until the agreement is complete.
TABLES.
TABLE I. (See articles 42, 45.)
ELLIPSE.
HYPERBOLA.
A
LogB
C
T
LogB
C
T
0.000
0
0
0.00000
0
0
0.00000
.001
0
0
.00100
0
0
.00100
.002
0
2
.00200
0
2
.00200
.003
1
4
.00301
1
4
.00299
.004
1
7
.00401
1
7
.00399
0.005
2
11
0.00502
2
11
0.00498
.006
3
16
.00603
3
16
.00597
.007
4
22
.00704
4
22
.00696
.008
5
29
.00805
5
29
.00795
.009
6
37
.00907
6
37
.00894
0.010
7
46
0.01008
7
46
0.00992
.011
9
56
.01110
9
55
.01090
.012
11
66
.01212
11
66
.01189
.013
13
78
.01314
13
77
.01287
.014
15
90
.01416
15
89
.01384
0.015
17
103
0.01518
17
102
0.01482
.016
19
118
.01621
1 19
116
.01580
.017
22
133
.01723
21
131
.01677
.018
24
149
.01826
24
147
.01774
.019
27
166
.01929
27
164
.01872
0.020
30
184
0.02032
30
182
0.01968
.021
33
203
.02136
33
200
.02065
,022
36
223
.02239
36
220
.02162
.023
40
244
.02343
39
240
.02258
.024
43
265
.02447
43
261
.02355
0.025
47
288
0.02551
46
283
0.02451
.026
51
312
.02655
50
306
.02547
.027
55
336
.02760
54
330
.02643
.028
59
362
.02864
58
355
.02739
.029
63
388
.02969
62
381
.02834
0.030
67
416
0.03074
67
407
0.02930
.031
72
444
.03179
71
435
.03025
.032
77
473
.03284
76
463
.03120
.033
82
503
.03389
80
492
.03215
.034
87
535
.03495
85
523
.03310
0.035
92
567
0.03601
91
554
0.03404
.036
97
600
.03707
96
585
.03499
.037
103
634
.03813
101
618
.03593
.038
108
669
.03919
107
652
.03688
.039
114
704
.04025
112
686
.03782
.040
120
741
.04132
118
722
.03876
TABLE
ELLIPSE.
HYPERBOLA. 1
A
LogB
C
T
LogB
c
T
0.040
120
741
0.041319
118
722
0.038757
.041
126
779
.042387
124
758
.039695
.042
133
818
.043457
130
795
.040632
.043
139
858
.044528
136
833
.041567
.044
146
898
.045601
143
872
.042500
0.045
152
940
0.046676
149
912
0.043432
.046
159
982
.047753
156
953
.044363
.047
166
1026
.048831
163
994
.045292
.048
173
1070
.049911
170
1037
.046220
.049
181
1116
.050993
177
1080
.047147
0.050
188
1162
0.052077
184
1124
0.048072
.051
196
1210
.053163
191
1169
.048995
.052
204
1258
.054250
199
1215
.049917
.053
212
1307
.055339
207
1262
.050838
.054
220
1358
.056430
215
1310
.051757
0.055
228
1409
0.057523
223
1358
0.052675
.056
286
1461
.058618
231
1407
.053592
.057
245
1514
.059714
239
1458
.054507
.058
254
1568
.060812
247
1509
.055420
.059
263
1623
.061912
25 G
1561
.056332
0.060
272
1679
0.063014
265
1614
0.057243
.061
281
1736
.064118
273
1667
.058152
.062
290
1794
.065223
282
1722
.059060
.063
300
1853
.066331
291
1777
.059967
.064
309
1913
.067440
301
1833
.060872
0.065
319
1974
0.068551
310
1891
0.061776
.066
329
2036
.069664
320
1949
.062678
.067
339
2099
.070779
329
2007
.063579
.068
350
2163
.071896
339
2067
.064479
.069
360
2228
.073014
349
2128
.065377
0.070
371
2294
0.074135
359
2189
0.066274
.071
381
2360
.075257
370
2251
.067170
.072
392
2428
.076381
380
2314
.068064
.073
403
2497
.077507
390
2378
.068957
.074
415
2567
.078635
401
2443
.069848
0.075
426
2638
0.079765
412
2509
0.070738
.076
437
2709
.080897
423
2575
.071627
.077
449
2782
.082030
434
2643
.072514
.078
461
2856
.083166
445
2711
.073400
.079
473
2930
.084303
457
2780
.074285
.080
485
3006
.085443
468
2850
.075168
TABLE I,
ELLIPSE.
HYPERBOLA.
A
LogB
C
T
LogB
C
T
0.080
485
3006
0.085443
468
2850
0.075168
.081
498
3083
.086584
480
2921
.076050
.082
510
3160
.087727
492
2992
.076930
.083
523
3239
.088872
504
3065
.077810
.084
535
3319
.090019
516
3138
.078688
0.085
548
3399
0.091168
528
3212
0.079564
.086
561
3481
.092319
540
3287
.080439
.087
575
3564
.093472
553
3363
.081313
.088
588
3647
.094627
566
3440
.082186
.089
602
3732
.095784
578
3517
.083057
0.090
615
3818
0.096943
591
3595
0.083927
.091
629
3904
.098104
604
3674
.084796
.092
643
3992
.099266
618
3754
.085663
.093
658
4081
.100431
631
3835
.086529
.094
672
4170
.101598
645
3917
.087394
0.095
687
4261
0.102766
658
3999
0.088257
.096
701
4353
.103937
672
4083
.089119
.097
716
4446
.105110
686
4167
.089980
.098
731
4539
.106284
700
4252
.090840
.099
746
4634
.107461
714
4338
.091698
0.100
762
4730
0.108640
728
4424
0.092555
.101
777
4826
.109820
743
4512
.093410
.102
793
4924
.111003
758
4600
.094265
.103
809
5023
.112188
772
4689
.095118
.104
825
5123
.113375
787
4779
.095969
0.105
841
5224
0.114563
802
4870
0.096820
.106
857
5325
.115754
817
4962
.097669
.107
873
5428
.116947
833
5054
.098517
.108
890
5532
.118142
848
5148
.099364
.109
907
5637
.119339
864
5242
.100209
0.110
924
5743
0.120538
880
5337
0.101053
•111
941
5850
.121739
895
5432
.101896
.112
958
5958
.122942
911
5529
.102738
.113
975
6067
.124148
928
5626
.103578
.114
993
6177
.125355
944
5724
.104417
0.115
1011
6288
0.126564
960
5823
0.105255
.116
1029
6400
.127776
977
5923
.106092
.117
1047
6513
.128989
994
6024
.106927
.118
1065
6627
.130205
1010
6125
.107761
.119
1083
6742
.131423
1027
6228
.108594
.120
1102
6858
.132643
1045
G331
.109426
1
TABLE 1,
ELLIPSE.
HYPERBOLA. 1
A
LogB
C
T
LogB
C
T
0.120
1102
6858
0.132643
1045
6331
0.109426
.121
1121
6976
.133865
1062
6435
.110256
.122
1139
7094
.135089
107-9
6539
.111085
.123
1158
7213
.136315
1097
6645
.111913
.124
1178
7334
.137543
1114
6751
.112740
0.125
1197
7455
0.138774
1132
6858
0.113566
.126
1217
7577
.140007
1150
6966
.114390
.127
1236
7701
.141241
1168
7075
.115213
.128
1256
7825
.142478
1186
7185
.116035
.129
1276
7951
.143717
1205
7295
.116855
0.130
1296
8077
0.144959
1223
7406
0.117075
.131
1317
8205
.146202
1242
7518
.118493
.132
1337
8334
.147448
1261
7631
.119310
.133
1358
8463
.148695
1280
7745
.120126
.134
1378
8594
.149945
1299
7859
.120940
0.135
1399
8726
0.151197
1318
7974
0.121754
.136
1421
8859
.152452
1337
8090
.122566
.137
1442
8993
.153708
1357
8207
.123377
.138
1463
9128
.154967
1376
8325
.124186
.139
1485
9264
.156228
1396
8443
.124995
0.140
1507
9401
0.157491
1416
8562
0.125802
.141
1529
9539
.158756
1436
8682
.126609
.142
1551
9678
.160024
1456
8803
.127414
.143
1573
9819
.161294
1476
8925
.128217
.144
1596
9960
.162566
1497
9047
.129020
0.145
1618
10102
0.163840
1517
9170
0.129822
.146
1641
10246
.165116
1538
9294
.130622
.147
1664
10390
.166395
1559
9419
.131421
.148
1687
10536
.167676
1580
9545
.132219
.149
1710
10683
.168959
1601
9671
.133016
0.150
1734
10830
0.170245
1622
9798
0.133812
.151
1757
10979
.171533
1643
9926
.134606
.152
1781
11129
.172823
1665
10055
.135399
.153
1805
11280
.174115
1686
10185
.136191
.154
1829
11432
.175410
1708
10315
.136982
0.155
1854
11585
0.176707
1730
10446
0.137772
.156
1878
11739
.178006
1752
10578
.138561
.157
1903
11894
.179308
1774
10711
.139349
.158
1927
12051
.180612
1797
10844
.140135
.159
1952
12208
.181918
1819
10978
.140920
.160
1977
12366
.183226
1842
11113
.141704
TABLE I.
ELLIPSE.
HYPEEBOLA.
A
LogB
C
T
LogB
C
T
0.160
1977
12366
0.183226
1842
11113
0.141704
.161
2003
12526
.184537
1864
11249
.142487
.162
2028
12686
.185850
1887
11386
.143269
.163
2054
12848
.187166
1910
11523
.144050
.164
2080
13011
.188484
1933
11661
.144829
0.165
2106
13175
0.189804
1956
11800
0.145608
.166
2132
13340
.191127
1980
11940
.146385
.167
2158
13506
.192452
2003
12081
.147161
.168
2184
13673
.193779
2027
12222
.147937
.169
2211
13841
.195109
2051
12364
.148710
0.170
2238
14010
0.196441
2075
12507
0.149483
.171
2265
14181
.197775
2099
12651
.150255
.172
2292
14352
.199112
2123
12795
.151026
.173
2319
14525
.200451
2147
12940
.151795
.174
2347
14699
.201793
2172
13086
.152564 ,
0.175
2374
14873
0.203137
2196
13233
0.153331
.176
2402
15049
.204484
2221
13380
.154097
.177
2430
15226
.205832
2246
13529
.154862
.178
2458
15404
.207184
2271
13678
.155626
.179
2486
15583
.208538
2296
13827
.156389
0.180
2515
15764
0.209894
2321
13978
0.157151
.181
2543
15945
.211253
2346
14129
.157911
.182
2572
16128
.212614
2372
14281
.158671
.183
2601
16311
.213977
2398
14434
.159429
.184
2630
16496
.215343
2423
14588
.160187
0.185
2660
16682
0.216712
2449
14742
0.160943
.186
2689
16868
.218083
2475
14898
.161698
.187
2719
17057
.219456
2502
15054
.162453
.188
2749
17246
.220832
2528
15210
.163206
.189
2779
17436
.222211
2554
15368
.163958
0.190
2809
17627
0.223592
2581
15526
0.164709
.191
2839
17820
.224975
2608
15685
.165458
.192
2870
18013
.226361
2634
15845
.166207
.193
2900
18208
.227750
. 2661
16005
.166955
.194
2931
18404
.229141
2688
16167
.167702
0.195
2962
18601
0.230535
2716
16329
0.168447
.196
2993
18799
.231931
2743
16491
.169192
.197
3025
18998
.233329
2771
16655
.169935
.198
3056
19198
.234731
2798
16819
.170678
.199
3088
19400
.236135
2826
16984
.171419
.200
3120
19602
.237541
2854
17150
.172159
TABLE I.
ELLIPSE.
HYPERBOLA.
A
LogB
C
T
LogB
C
T
0.200
3120
19602
0.237541
2854
17150 (
).172159
.201
3152
19806
.238950
2882
17317
.172899
.202
3184
200U
.240361
2910
17484
.173637
.203
3216
20217
.241776
2938
17652
.174374
.204
3249
20424
.243192
2967
17821
.175110
0.205
3282
20632
0.244612
2995
17991 (
).l 75845
.206
3315
20842
.246034
3024
181G1
.176579
.207
3348
21052
.247458
3053
18332
.177312
.208
3381
21264
.248885
3082
18504
.178044
.209
3414
21477
.250315
3111
18677
.178775
0.210
3448
21690
0.251748
3140
18850 (
).179505
.211
3482
21905
.253183
31G9
19024
.180234
.212
3516
22122
.254620
3199
19199
.180962
.213
3550
22339
.256061
3228
19375
.181688
.214
3584
22557
.257504
3258
19551
.182414
0.215
3618
22777
0.258950
3288
19728 (
).183139
.216
3653
22998
.260398
3318
19906
.183863
.21,7
3688
23220
.261849
3348
20084
.184585
.218
3723
23443
.263303
3378
20264
.185307
.219
3758
23667
.264759
3409
20444
.186028
0.220
3793
23892
0.266218
3439
20625 (
).186747
.221
3829
24119
.267680
3470
20806
.187466
.222
3865
24347
.269145
3500
20988
.188184
.223
3900
24576
.270612
3531
21172
.188900
.224
3936
24806
.272082
3562
21355
.189616
0.225
3973
25037
0.273555
3594
21540 (
).l 90331
.226
4009
25269
.275031
3625
21725
.191044
.227
4046
25502
.276509
3656
21911
.191757
.228
4082
25737
.277990
3688
22098
.192468
.229
4119
25973
.279474
3719
22285
.193179
0.230
4156
26210
0.280960
3751
22473 (
).l 93889
.231
4194
26448
.282450
3783
22662
.194597
.232
4231
26687
.283942
3815
22852
.195305
.233
4269
26928
.285437
3847
23042
.196012
.234
4306
27169
.286935
3880
23234
.196717
0.235
4344
27412
0.288435
3912
23425 (
).197422
.236
4382
27656
.289939
3945
23618
.198126
.237
4421
27901
.291445
3977
23811
.198829
.238
4459
28148 '
.292954
4010
24005
.199530
.239
4498
28395
.294466
4043
24200
.200231
.210
4537
28644
.295980
4076
24396
.200931
TABLE I,
ELLIPSE.
HYPERBOLA.
A
LogB
C
T
LogB
C
T
0.240
4537
28644
0.295980
4076
24396
0.200931
.241
4576
28894
.297498
4110
24592
.201630
.242
4615
29145
.299018
4143
24789
.202328
.243
4654
29397
.300542
4176
24987
.203025
.244
4694
29651
.302068
4210
25185
.203721
0.245
4734
29905
0.303597
4244
25384
0.204416
.246
4774
30161
.305129
4277
25584
.205110
.247
4814
30418
.306664
4311
25785
.205803
.248
4854
30676
.308202
4346
25986
.206495
.249
4894
30935
.309743
4380
26188
.207186
0.250
4935
31196
0.311286
4414
26391
0.207876
.251
4976
31458
.312833
4449
26594
.208565
.252
5017
31721
.314382
4483
26799
.209254
.253
5058
31985
.315935
4518
27004
.209941
.254
5099
32250
.317490
4553
27209
.210627
0.255
5141
32517
0.319048
4588
27416
0.211313
.256
5182
32784
.320610
4623
27623
.211997
.257
5224
33053
.322174
4658
27830
.212681
.258
5266
33323
.323741
4694
28039
.213364
.259
5309
33595
.325312
*
4729
28248
.214045
0.260
5351
33867
0.326885
4765
28458
0.214726
.261
5394
34141
,328461
4801
28669
.215406
.262
5436
34416
.330041
4838
28880
.216085
.263
5479
34692
.331623
4873
29092
.216763
.264
5522
34970
.333208
4909
29305
.217440
0.265
5566
35248
0.334797
4945
29519
0.218116
.266
5609
35528
.336388
4981
29733
.218791
.267
5653
35809
.337983
5018
29948
.219465
.268
5697
36091
.339580
5055
30164
.220138
.269
5741
36375
.341181
5091
30380
.220811
0.270
5785
36659
0.342785
5128
30597
0.221482
.271
5829
36945
.344392
5165
30815
.222153
.272
5874
37232
.346002
5202
31033
.222822
.273
5919
37521
.347615
5240
31253
.223491
.274
5964
37810
.349231
5277
31473
.224159
0.275
6009
38101
0.350850
5315
31693
0.224826
.276
6054
38393
.352473
5352
31915
.225492
.277
6100
38686
.354098
5390
32137
.226157
.278
6145
38981
.355727
5428
32359
.226821
.279
6191
39277
.357359
5466
32583
.227484
.280
6237
39573
.358994
5504
32807
.228147
TABLE I,
ELLIPSE.
HYPERBOLA.
A
LogB
c
T
LogB
5504
C
T
0.280
G237
39573
0.358994
32807
0.228147
.281
6283
39872
.360632
5542
33032
.228808 1
.282
6330
40171
.362274
5581
33257
.229469
.283
6376
40472
.363918
5619
33484
.230128 !
.284
6423
40774
.365566
5658
33711
.230787
0.285
6470
41077
0.367217
5697
33938
0.231445
.286
6517
41381
.368871
5736
34167
.232102
.287
6564
41687
.370529
5775
34396
.232758
.288
6612
41994
.372189
5814
34626
.233413
.289
6660
42302
.373853
5853
34856
.234068
0.290
6708
42611
0.375521
5893
35087
0.234721
.291
6756
42922
.377191
5032
35319
.235374
.292
6804
43233
. .378865
5972
35552
.236025
.293
6852
43547
.380542
6012
35785
.236676
.294
6901
43861
.382222
6052
36019
.237326
0.295
6950
44177
0.383906
6092
36253
0.237975
.296
6999
44493
.385593
6132
36489
.238623
-.297
7048
44812
.387283
6172
36725
.239271
.298
7097
45131
.388977
6213
36961
.239917
.299
7147
45452
.390673
6253
37199
.240563
j .300
7196
45774
.392374
6294
37437
.241207
TABLE 11. (See Article 93.)
h
logyy
h
logyy
h
logyy
0.0000
0.0000000
0.0040
0.0038332
0.0080
0.0076133
.0001
.0000965
.0041
.0039284
.0081
.0077071
.0002
.0001930
.0042
.0040235
.0082
.0078009
.0003
.0002894
.0043
.0041186
.0083
.0078947
.0004
.0003858
.0044
.0042136
.0084
.0079884
0.0005
0.0004821
0.0045
0.0043086
0.0085
0.0080821
.0006
.0005784
.0046
.0044036
.0086
.0081758
.0007
.0006747
.0047
.0044985
.0087
.0082694
.0008
.0007710
.0048
.0045934
.0088
.0083630
.0009
.0008672
.0049
.0046883
.0089
.0084566
0.0010
0.0009634
0.0050
0.0047832
0.0090
0.0085502
, .0011
.0010595
.0051
.0048780
.0091
.0086437
.0012
.0011556
.0052
.0049728
.0092
.0087372
.0013
, .0012517
.0053
.0050675
.0093
.0088306
.0014
.0013478
.0054
.0051622
.0094
.0089240
0.0015
0.0014438
0.0055
0.0052569
0.0095
0.0090174
.0016
.0015398
.0056
.0053515
.0096
.0091108
.0017
.0016357
.0057
.0054462
.0097
.0092041
.0018
.0017316
.0058
.0055407
.0098
.0092974
.!.)19
.0018275
.0059
.0056353
.0099
.0093906
0.0020
0.0019234
0.0060
0.0057298
0.0100
0.0094838
.0021
.0020192
.0061
.0058243
.0101
.0095770
.0022
.0021150
.0062
.0059187
.0102
.0096702
.0023
.0022107
.0063
.0060131
.0103
.0097633
.0024
.0023064
.0064
.0061075
.0104
.0098564
0.0025
0.0024021
0.0065
0.0062019
0.0105
0.0099495
.0026
.0024977
.0066
.0062962
.0106
.0100425
.0027
.0025933
.0067
.0063905
.0107
.0101355
.0028
.0026889
.0068
.0064847
.0108
.0102285
.0029
.0027845
.0069
.0065790
.0109
.0103215
0.0030
0.0028800
0.0070
0.0066732
0.0110
0.0104144
.0031
.0029755
.0071
.0067673
.0111
.0105073
.0032
.0030709
.0072
.0068614
.0112
.0106001
.0033
.0031663
.0073
.0069555
.0113
.0106929
.0034
.0032617
.0074
.0070496
.0114
.0107857 1
0.0035
0.0033570
0.0075
0.0071436
0.0115
i
0.0108785 1
.0036
.0034523
.0076
.0072376
.0116
.0109712
.0037
.0035476
.0077
.0073316
.0117
.0110639
.0038
.0036428
.0078
.0074255
.0118
.0111565
.0039
.0037380
.0079
.0075194
.0119
.0112491
.0040
.0038332
.0080
.0076133
.0120
.0113417
10
TABLE II.
h
logyy
h
logyy
h
logyy
0.0120
0.0113417
0.0160
0.0150202
0.0200
0.0186501
.0121
.0114343
.0161
.0151115
.0201
.0187403
.0122
.0115268
.0162
.0152028
.0202
.0188304
.0123
.0116193
.0163
.0152941
.0203
.0189205
.0124
.0117118
.0164
.0153854
.0204
.0190105
0.0125
0.0118043
0.0165
0.0154766
0.0205
0.0191005
.0126
.0118967
.0166
.0155678
.0206
.0191905
.0127
.0119890
.0167
.0156589
.0207
.0192805
.0128
.0120814
.0168
.0157500
.0208
.0193704
.0129
.0121737
.0169
.0158411
.0209
.0194603
0.0130
0.0122660
0.0170
0.0159322
0.0210
0.0195502
.0131
.0123582
.0171
.0160232
.0211
.0196401
.0132
.0124505
.0172
.0161142
.0212
.0197299
.0133
.0125427
.0173
.0162052
.0213
.0198197
.0134
.0126348
.0174
.0162961
.0214
.0199094
0.0135
0.0127269
0.0175
0.0163870
0.0215
0.0199992
.0136
.0128190
.0176
.0164779
.0216
.0200889
.0137
.0129111
.0177
.0165688
.0217
.0201785
.0138
.0130032
.0178
.0166596
.0218
.0202682
.0139
.0130952
.0179
.0167504
.0219
.020357.S
0.0140
0.0131871
0.0180
0.0168412
0.0220
0.0204474
.0141
.0132791
.0181
.0169319
.0221
.0205369
.0142
.0133710
.0182
.0170226
.0222
.0206264
.0143
.0134629
.0183
.0171133
.0223
.0207159
.0144
.0135547
.0184
.0172039
.0224
.0208054
0.0145
0.0136465
0.0185
0.0172945
0.0225
0.0208948
.0146
.0137383
.0186
.0173851
.0226
.0209842
.0147
.0138301
.0187
.0174757
.0227
.0210736
.0148
.0139218
.0188
.0175662
.0228
.0211630
.0149
.0140135
.0189
.0176567
.0229
.0212523
0.01,50
0.0141052
0.0190
0.0177471
0.0230
0.0213416
.0151
.0141968
.0191
.0178376
.0231
.0214309
.0152
.0142884
.0192
.0179280
.0232
.0215201
.0153
.0143800
.0193
.0180183
.0233
• .0216093
.0154
.0144716
.0194
.0181087
.0234
.0216985
0.0155
0.0145631
0.0195
0.0181990
0.0235
0.0217876
.0156
.0146546
.0196
.0182893
.0236
.0218768
.0157
.0147460
.0197
.0183796
.02;!7
.0219659
.0158
.0148374
.0198
.0184698
.0238
.0220549
.0159
.0149288
.0199
.0185600
.0239
.0221440
.0160
.0150202
.0200
.0186501
.0240
.0222330
1
TABLE II.
11
h
logyy
h
logyy
h
logyy
0.0240
0.0222330
0.0280
0.0257700
0.0320
0.0292626
.0241
.0223220
. .0281
.0258579
.0321
.0293494
.0242
.0224109
.0282
.0259457
.0322
.0294361
.0243
.0224998
.0283
.0260335
.0323
.0295228
.0244
.0225887
.0284
.0261213
.0324
.0296095
0.0245
0.0226776
0.0285
0.0262090
0.0325
0.0296961
.0246
.0227664
.0286
.0262967
.0326
.0297827
.0247
.0228552
.0287
.0263844
.0327
.0298693
.0248
.0229440
.0288
.0264721
.0328
.0299559
.0249
.0230328
.0289
.0265597
.0329
.0300424
0.0250
0.0231215
0.0290
0.0266473
0.0330
0.0301290
.0251
.0232102
.0291
.0267349
.0331
.0302154
.0252
.0232988
.0292
.0268224
.0332
.0303019
.0253
.0233875
.0293
.0269099
.0333
.0303883
.0254
.0234761
.0294
.0269974
.0334
.0304747
0.0255
0.0235647
0.0295
0.0270849
0.0335
0.0305611
.0256
.0236532
.0296
.0271723
.0336
.0306475
.0257
.0237417
.0297
.0272597
.0337
.0307338
.0258
.0238302
.0298
.0273471
.0338
.0308201
.0259
.0239187
.0299
.0274345
.0339
.0309064
0.0260
0.0240071
0.0300
0.0275218
0.0340
0.0309926
.0261
.0240956
.0301
.0276091
.0341
.0310788
.0262
.0241839
.0302
.0276964
.0342
.0311650
.0263
.0242723
.0303
.0277836
.0343
.0312512
.0264
.0243606
.0304
.0278708
.0344
.0313373
0.0265
0.0244489
0.0305
0.0279580
0.0345
0.0314234
.0266
.0245372
.0306
.0280452
.0346
.0315095
.0267
.0246254
.0307
.0281323
.0347
.0315956
.0268
.0247136
.0308
.0282194
.0348
.(1316816
.0269
.0248018
.0309
.0283065
.0349
.0317676
0.0270
0.0248900
0.0310
0.0283936
0.0350
0.0318536
.0271
.0249781
.0311
.0284806
.0351
.0319396
.0272
.0250662
.0312
.0285676
.0352
.0320255
.0273
.0251543
.0313
.0286546
.0353
.0321114
.0274
.0252423 .
.0314
.0287415
.0354
.0321973
0.0275
0.0253303
0.0315
0.0288284
0.0355
0.0322831
.0276
.0254183
.0316
.0289153
.0356
.0323689
.0277
.0255063
.0317
.0290022
.0357
.0324547
.0278
.0255942
.0318
.0290890
.0358
.0325405
.0279
.0256821
.0319
.0291758
.0359
.0326262
.0280
.0257700
.0320
.0292626
.0360
.0327120
12
TABLE II.
h
logyy
h
logyy
h
logyy
0.0360
3.0327120
0.040
0.0361192
0.080
0.0681057
.0361
.0327976
.041
.0369646
.081
.0688612
.0362
.0328833
.042
.0378075
.082
.0696146
.0363
.0329689
.043
.0386478
.083
.0703661
.0304
.0330546
.044
.0394856
.084
.0711157
0.0365 (
).0331401
0.045
0.0403209
0.085
0.0718633
.0366
.0332257
.046
.0411537
.086
.0726090
.0367
.0333112
.047
.0419841
.087
.0733527
.0368
.0333967
.048
.0428121
.088
.0740945
.0369
.0334822
.049
.0436376
.089
.0748345
0.0370 (
).0335677
0.050
0.0444607
0.090
0.0755725
.0371
.0336531
.051
.0452814
.091
.0763087
.0372
.0337385
.052
.0460997
.092
.0770430
.0373
.0338239
.053
.0469157
.093
.0777754
.0374
.0339092
.054
.0477294
.094
.0785060
0.0375 (
).0339946
0.055
0.0485407
0.095
0.0792348
.0376
.0340799
.056
.0493496
.096
.0799617
.0377
.0341651
.057
.0501563
.097
.0806868
.0378
.0342504
.058
.0509607
.098
.0814101
.0379
.0343356
.059
.0517628
.099
.0821316
0.0380 (
).0344208
0.060
0.0525626
0.100
0.0828513
.0381
.0345059
.061
.0533602
.101
.0835693
.0382
.0345911
.062
.0541556
.102
.0842854
.0383
.0346762
.063
.0549488
.103
.0849999
.0384
.0347613
.064
.0557397
.104
.0857125
0.0385 (
).0348464
0.065
0.0565285
0.105
0.0864235
.0386
.0349314
.066
.0573150
.106
.0871327
.0387
.0350164
.067
.0580994
.107
.0878401
.0388
.0351014
.068
.0588817
.108
.0885459
.0389
.0351864
.069
.0596618
.109
.0892500
0.0390 (
).0352713
0.070
0.0604398
0.110
0.0899523
.0391
.0353562
.071
.0612157
.111
.0906530
.0392
.0354411
.072
.0619895
.112
.091.3520
.0393
.0355259
.073
.0627612
.113
.0920494
.0394
.0356108
.074
.0635308
.114
.0927451
0.0395 (
).0356956
0.075
0.0642984
0.115
0.0934391
.0396
.0357804
.076
.0650639
.116
.0941315
.0397
.0358651
.077
.0658274
.117
.0948223
.0398
.0359499
.078
.0665888
.118
.0955114
.0399
.0360346
.079
.0673483
.119
.0961990
.0400
.0361192
.080
.0681057
.120
.0968849
TABLE II.
h
logyy
h
logyy
h
logyy
0.120
0.0968849
0.160
0.1230927
0.200
0.1471869
.121
.0975692
.161
.1237192
.201
.1477653
.122
.0982520
.162
.1243444
.202
.1483427
.123
.0989331
.163
.1249682
.203
.1489189
.124
.0996127
.164
.1255908
.204
.1494940
0.125
0.1002907
0.165
0.1262121
0.205
0.1500681
.126
.1009672
.166
.1268321
.206
.1506411
.127
.1016421
.167
.1274508
.207
.1512130
.128
.1023154
.168
.1280683
.208
.1517838
.129
.1029873
.169
.1286845
.209
.1523535
0.130
0.1036576
0.170
0.1292994
0.210
0.1529222
.131
.1043264
.171
.1299131
.211
.1534899
.132
.1049936
.172
.1305255
.212
.1540565
.133
.1056594
.173
.1311367
.213
.1546220
.134
.1063237
.174
.1317466
.214
.1551865
0.135
0.1069865
0.175
0.1323553
0.215
0.1557499
.136
.1076478
.176
.1329628
.216
.1563123
.137
.1083076
.177
.1335690
.217
.1568737
.138
.1089660
.178
.1341740
.218
.1574340
.139
.1096229
.179
.1347778
.219
.1579933
0.140
0.1102783
0.180
0.1353804
0.220
0.1585516
.141
.1109323
.181
.1359818
.221
.1591089
.142
.1115849
.182
.1365821
.222
.1596652
.143
.1122360
.183
.1371811
.223
.1602204
.144
.1128857
.184
.1377789
.224
.1607747
0.145
0.1135340
0.185
0.1383755
0.225
0.1613279
.146
.1141809
.186
.1389710
.226
.1618802
.147
.1148264
.187
.1395653
.227
.1624315
.148
.1154704
.188
.1401585
.228
.1629817
.149
.1161131
.189
.1407504
.229
.1635310
0.150
0.1167544
0.190
0.1413412
0.230
0.1640793
.151
.1173943
.191
.1419309
.231
.1646267
.152
.1180329
.192
.1425194
.232
.1651730
.153
.1186701
.193
.1431068
.233
.1657184
.154
.1193059
.194
.1436931
.234
.1662628
0.155
0.1199404
0.195
0.1442782
0.235
0.1668063
.156
.1205735
.196
.1448622
.236
.1673488
.157
.1212053
.197
.1454450
.237
.1678903
.158
.1218357
.198
.1460268
.238
.1684309
.159
.1224649
.199
.1466074
.239
.1689705
.160
.1230927
.200
.1471869
.240
.1695092
14
TABLE II.
h
logyy
h
logyy
h
logyy
0.240
0.1695092
0.280
0.1903220
0.320
0.2098315
.241
.1700470
.281
.1908249
.321
.2103040
.242
.1705838
.282
.1913269
.322
.2107759
.243
.1711197
.283
.1918281
.323
.2112470
.244
.1716547
.284
.1923286
.324
.2117174
0.245
0.1721887
0.285
0.1928282
0.325
0.2121871
.246
.1727218
.286
.1933271
.326
.2126562
.247
.1732540
.287
.1938251
.327
.2131245
.248
.1737853
.288
.1943224
.328
.2135921
.249
.1743156
.289
.1948188
.329
.2140591
0.250
0.1748451
0.290
0.1953145
0.330
0.2145253
.251
.1753736
.291
.1958094
.331
.2149909
.252
.1759013
.292
.1963035
.332
.2154558
.253
.1764280
.293
.1967968
.333
.2159200
.254
.1769538
.294
.1972894
.334
.2163835
0.255
0.1774788
0.295
0.1977811
0.335
0.2168464
.256
.1780029
.296
.1982721
..336
.2173085
.257
.1785261
.297
.1987624
.337
.2177700
.258
.1790484
.298
.1992518
.338
.2182308
.259
.1795698
.299
.1997406
.339
.2186910
0.260
0.1800903
0.300
0.2002285
0.340
0.2191505
.261
.1806100
.301
.2007157
.341
.2196093
.262
.1811288
.302
.2012021
.342
.2200675
.263
.1816467
.303
.2016878
.343
.2205250
.264
.1821638
.304
.2021727
.344
.2209818
0.265
0.1826800
0.305
0.2026569
0.345
0.2214380
.266
.1831953
.306
.2031403
.346
.2218935
.267
.1837098
.307
.2036230
.347
.2223483
.268
.1842235
.308
.2041050
.348
.2228025
.269
.1847363
.309
.2045862
.349
.2232561
0.270
0.1852483
0.310
0.2050667
0.350
0.2237090
.271
.1857594
.311
.2055464
.351
.2241613
.272
.1862696
.312
.2060254
.352
.2246130
.273
.1867791
.313
.2065037
.353
.2250640
.274
.1872877
.314
.2069813
.354
.2255143
0.275
0.1877955
0.315
0.2074581
0.355
0.2259640
.276
.1883024
.316
.2079342
.356
.2264131
.277
.1888085
.317
.2084096
.357
.2268615
.278
.1893138
.318
.2088843
.358
.2273093
.279
.1898183
.319
.2093582
.359
.2277565
.280
.1903220
.320
.2098315
.360
.2282031
TABLE II.
15
h
logyy
h
logyy
h
logyy
0.360
0.2282031
0.400
0.2455716
0.440
0.2620486
.361
.2286490
.401
.2459940
.441
.2624499
.862
.2290943
.402
.2464158
.442
.2628507
.363
.2295390
.403
.2468371
.443
.2632511
.364
.2299831
.404
.2472578
.444
.2636509
0.365
0.2304265
0.405
0.2476779
0.445
0.2640503
.366
.2308694
.406
.2480975
.446
.2644492
.367
.2313116
.407
.2485166
.447
.2648475
.368
.2317532
.408
.2489351
.448
.2652454
.369
.2321942
.409
.2493531
.449
.2656428
0.370
0.2326346
0.410
0.2497705
0.450
0.2660397
.371
.2330743
.411
.2501874
.451
.2664362
.372 ■
.2335135
.412
.2506038
.452
.2668321
.373
.2339521
.413
.2510196
.453
.2672276
.374
.2343900
.414
.2514349
.454
.2676226
0.375
0.2348274
0.415
0.2518496
0.455
0.2680171
.376
.2352642
.416
.2522638
.456
.2684111
.377
.2357003
.417
.2526775
.457
.2688046
.378
.2361359
.418
.2530906
.458
.2691977
.379
.2365709
.419
.2535032
.459
.2695903
0.380
0.2370053
0.420
0.2539153
0.460
0.2699824
.381
.2374391
.421
.2543269
.461
.2703741
.382
.2378723
.422
.2547379
.462
.2707652
.383
.2383050
.423
.2551485
.463
.2711559
.384
.2387370
.424
.2555584
.464
.2715462
0.385
0.2391685
0.425
0.2559679
0.465
0.2719360
.386
.2395993
.426
.2563769
.466
.2723253
.387
.2400296
.427
.2567853
.467
.2727141
.388
.2404594
.428
.2571932
.468
.2731025
.389
.2408885
.429
.2576006
.469
.2734904
0.390
0.2413171
0.430
0.2580075
0.470
0.2738778
.391
.2417451
.431
.2584139
.471
.2742648
.392
.2421725
.432
.2588198
.472
.2746513
.393
.2425994
.433
.2592252
.473
.2750374
.394
.2430257
.434
.2596300
.474
.2754230
0.395
0.2434514
0.435
0.2600344
0.475
0.2758082
.396
.2438766
.436
.2604382
.476
.2761929
.397
.2443012
.437
.2608415
.477
.2765771
.398
.2447252
.438
.2612444
.478
.2769609
.399
.2451487
.439
.2616467
.479
.2773443
.400
.2455716
.440
.2620486
.480
.2777272
16
TABLE II.
h.
logyy
h
logyy
h
logyy
0.480
0.2777272
0.520
0.2926864
0.560
0.3069938
.481
.2781096
.521
.2930518
.561
.3073437
.482
.2784916
.522
.2934168
.562
.3076931
.483
.2788732
.523
.2937813 •
.563
.3080422
.484
.2792543
.'524
.2941455
.564
.3083910
0.485
0.2796349
0.525
0.2945092
0.565
0.3087394
.486
.2800151
.526
.2948726
.566
.3090874
.487
.2803949
.527
.2952355
.567
.3094350
.488
.2807743
.528
.2955981
.568
.3097823
.489
.2811532
.529
.2959602
.569
.3101292
0.490
0.2815316
0.530
0.2963220
0.570
0.3104758
.491
.2819096
.531
.2966833
.571
.3108220
.492
.2822872
.532
.2970443
.572
.3111678
.493
.2826644
.533
.2974049
.573
.3115133
.494
.2830411
.534
.2977650
.574
.3118584
0.495
0.2834173
0.535
0.2981248
0.575
0.3122031
.496
.2837932
.536
.2984842
.576
.3125475
.497
.2841686
.537
.5988432
.577
.3128915
.498
.2845436
.538
.2992018
.578
.3132352
.499
.2849181
.539
.2995600
.579
.3135785
0.500
0.2852923
0.540
0.2999178
0.580
0.3139215
.501
.2856660
.541
.3002752
.581
.3142641
.502
.2860392
.542
.3006323
.582
.3146064
.503
.2864121
.543
.3009890
.583
.3149483
.504
.2867845
.544
.3013452
.584
.3152898
0.505
0.2871565
0.545
0.3017011
0.585
0.3156310
.506
.2875281
.546
.3020566
.586
.3159719
.507
.2878992
.547
.3024117
.587
.3163124
.508
.2882700
.548
.3027664
.588 '
.3166525
.509
.2886403
.549
.3031208
.589
.3169923
0.510
0.2890102
0.550
0.3034748
0.590
0.3173318
.511
.2893797
.551
.3038284
.591
.3176709
.512
.2897487
.552
.3041816
.592
.3180096
.513
.2901174
.553
.3045344
.593
.3183481
.514
.2904856
.554
.3048869
.594
.3186861
0.515
0.2908535
0.555
0.3052390
0.595
0.3190239
.516
.2912209
.556
.3055907
.596
.3193612
.517
.2915879
.557
.3059420
.597
.3196983
.518
.2919545
.558
.3062930
.598
.3200350
.519
.2923207
.559
.3066436
.599
.3203714
.520
.2926864
.560
.3069938
.600
.3207074
TABLE III. (See Articles 90, 100.)
17
xorz
S
f
X or z
f
f
0.000
0.0000000
0.0000000
0.040
0.0000936
0.0000894
.001
.0000001
.0000001
.041
.0000984
.0000938
.002
.0000002
.0000002
.042
.0001033
.0000984
.003
.0000005
.0000005
.043
.0001084
.0001031
.004
.0000009
.0000009
.044
.0801135
.0001079
0.005
0.0000014
0.0000014
0.045
0.0001188
0.0001128
.006
.0000021
.0000020
.046
.0001242
.0001178
.007
.0000028
.0000028
.047
.0001298
.0001229
.008
.0000037
.0000036
.048
.0001354
.0001281
.009
.0000047
.0000046
.049
.0001412
.0001334
0.010
0.0000058
0.0000057
0.050
0.0001471
0.0001389
.011
.0000070
.0000069
.051
.0001532
.0001444
.012
.0000083
.0000082
.052
.0001593
.0001500
.013
.0000097
.0000096
.053
.0001656
.0001558
.OU
.0000113
.0000111
.054
.0001720
.0001616
0.015
0.0000130
0.0000127
0.055
0.0001785
0.0001675
.016
.0000148
.0000145
.056
.0001852
.0001736
.017
.0000167
.0000164
.057
.0001920
.0001798
.018
.0000187
.0000183
.058
.0001989
.0001860
.019
.0000209
.0000204
.059
.0002060
.0001924
0.020
0.0000231
0.0000226
0.060
0.0002131
0.0001988
.021
.0000255
.0000249
.061
.0002204
.0002054
.022
.0000280
.0000273
.062
.0002278
.0002121
.023
.0000306
.0000298
.063
.0002354
.0002189
.024
.0000334
.0000325
.064
.0002431
.0002257
0.025
0.0000362
0.0000352
0.065
0.0002509
0.0002327
.026
.0000392
.0000381
.066
.0002588
.0002398
.027
.0000423
.0000410
.067
.0002669
.0002470
.028
.0000455
.0000441
.068
.0002751
.0002543
.029
.0000489
.0000473
.069
.0002834
.0002617
0.030
0.0000523
0.0000506
0.070
0.0002918
0.0002691
.031
.0000559
.0000539
.071
.0003004
.0002767
.032
.0000596
.0000575
.072
.0003091
.0002844
.033
.0000634
.0000611
.073
.0003180
.0002922
.034
.0000674
.0000648
.074
.0003269
.0003001
0.035
0.0000714
0.0000686
0.075
0.0003360
0.0003081
.036
.0000756
.0000726
.076
.0003453
.0003162
.037
.0000799
.0000766
.077
.0003546
.0003244
.038
.0000844
.0000807
.078
.0003641
.0003327
.039
.0000889
.0000850
.079
.0003738
.0003411
.040
.0000936
.0000894
.080
.0003835
.0003496
18
TABLE III.
X or z
f
^
X or z
f
f
0.080
0.0003835
0.0003496
0.120
0.0008845
0.0007698
.081
.0003934
.0003582
.121
.0008999
.0007822
.082
.0004034
.0003669
.122
.0009154
.0007948
.083
.0004136
.0003757
.123
.0009311
.0008074
.084
,0004239
.0003846
.124
.0009469
.0008202
0.085
0.0004343
0.0003936
0.125
0.0009628
0.0008330
.086
.0004448
.0004027
.126
.0009789
.0008459
.087
.0004555
.0004119
.127
.0009951
.0008590
.088
.0004663
.0004212
.128
.0010115
.0008721
.089
.0004773
.0004306
.129
.0010280
.0008853
0.090
0.0004884
0.0004401
0.130
0.0010447
0.0008986
.091
.0004996
.0004496
.131
.0010615
.0009120
.092
.0005109
.0004593
.132
.0010784
.0009255
.093
.0005224
.0004691
.133
.0010955
.0009390
.094
.0005341
.0004790
.134
.0011128
.0009527
0.095
0.0005458
0.0004890
0.135
0.0011301
0.0009665
.096
.0005577
.0004991
.136
.0011477
.0009803
.097
.0005697
.0005092
.137
.0011654
.0009943
.098
.0005819
.0005195
.138
.0011832
.0010083
.099
.0005942
.0005299
.139
.0012012
.0010224
0.100
0.0006066
0.0005403
0.140
0.0012193
0.0010366
.101
.0006192
.0005509
.141
.0012376
.0010509
.102
.0006319
.0005616
.142
.0012560
.0010653
.103
.0006448
.0005723
.143
.0012745
.0010798
.104
.0006578
.0005832
.144
.0012933
.0010944
0.105
0.0006709
0.0005941
0.145
0.0013121
0.0011091
.106
.0006842
.0006052
.146
.0013311
.0011238
.107
.0006976
.0006163
.147
.0013503
.0011387
.108
.0007111
.0006275
.148
.0013696
.0011536
.109
.0007248
.0006389
.149
.0013891
.0011686
0.110
0.0007386
0.0006503
0.150
0.0014087
0.0011838
.111
.0007526
.0006618
.151
.0014285
.0011990
.112
.0007667
.0006734
.152
.0014484
.0012143
.113
.0007809
.0006851
.153
.0014684
.0012296
.114
.0007953
.0006969
.154
.0014886
.0012451
0.115
0.0008098
0.0007088
0.155
0.0015090
0.0012607
.116
.0008245
.0007208
.156
.0015295
.0012763
.117
.0008393
.0007329
.157
.00] 5.502
.0012921
.118
.0008542
.0007451
.158
.0015710
.0013079
.119
.0008693
.0007574
.159
.0015920
.0013238
.120
.0008845
.0007698
.160
.0016131
.0013398
TABLE III.
19
X or z
f
f
X or z
^
f
0.160
0.0016131
0.0013398
0.200
0.0025877
0.0020507
.161
.0016344
.0013559
.201
.0026154
.0020702
.162
.0016559
.0013721
.202
.0026433
.0020897
.163
.0016775
.0013883
.203
.0026713
.0021094
.164
.0016992
.0014047
.204
.0026995
.0021292
0.165
0.0017211
0.0014211
0.205
0.0027278
0.0021490
.166
.0017432
.0014377
.206
.0027564
.0021689
.167
.0017654
.0014543
.207
.0027851
.0021889
.168
.0017878
.0014710
.208
.0028139
.0022090
.169
.0018103
.0014878
.209
.0028429
.0022291
0.170
0.0018330
0.0015047
0.210
0.0028722
0.0022494
.171
.0018558
.0015216
.211
.0029015
.0022697
.172
.0018788
.0015387
.212
.0029311
.0022901
.173
.0019020
.0015558
.213
.0029608
.0023106
.174
.0019253
.0015730
.214
.0029907
.0023311
0.175
0.0019487
0.0015903
0.215
0.0030207
0.0023518
.176
.0019724
.0016077
.216
.0030509
.0023725
.177
.0019961
.0016252
.217
.0030814
.0023932
.178
.0020201
.0016428
.218
.0031119
.0024142
.179
.0020442
.0016604
.219
.0031427
.0024352
0.180
0.0020685
0.0016782
0.220
0.0031736
0.0024562
.181
.0020929
.0016960
.221
.0032047
.0024774
.182
.0021175
.0017139
.222
.0032359
.0024986
.183
.0021422
.0017319
.223
.0032674
.0025199
.184
.0021671
.0017500
.224
.0032990
.0025412
0.185
0.0021922
0.0017681
0.225
0.0033308
0.0025627
.186
.0022174
.0017864
.226
.0033627
.0025842
.187
.0022428
.0018047
.227
.0033949
.0026058
.188
.0022683
.0018231
.228
.0034272
.0026275
.189
.0022941
.0018416
.229
.0034597
.0026493
0.190
0.0023199
0.0018602
0.230
0.0034924
0.0026711
.191
.0023460
.0018789
.231
.0035252
.0026931
.192
.0023722
.0018976
.232
.0035582
.0027151
.193
.0023985
.0019165
.233
.0035914
.0027371
.194
.0024251
.0019354
.234
.0036248
.0027593
0.195
0.0024518
0.0019544
0.235
0.0036584
0.0027816
.196
.0024786
.0019735
.236
.0036921
.0028039
.197
.0025056
.0019926
.237
.0037260
.0028263
.198
.0025328
.0020119
.238
.0037601
.0028487
.199
.0025602
.0020312
.239
.0037944
.0028713
.200
.0025877
.0020507
.240
.0038289
.0028939
20
TABLE III.
X or z
f
f
X or z
•f
f
0.240
0.0038289
0.0028939
0.270
0.0049485
0.0036087
.241
.0038635
.0029166
.271
.0049888
.0036337
.242
.0038983
.0029394
.272
.0050292
.0036587
.243
.0039333
.0029623
.273
.0050699
.0036839
.244
.0039685
.0029852
.274
.0051107
.0037091
0.245
0.0040039
0.0030083
0.275
0.0051517
0.0037344
.246
.0040394
.0030314
.276
.0051930
.0037598
.247
.0040752
.0030545
.277
.0052344
.0037852
.248
.0041111
.0030778
.278
.0052760
.0038107
.249
.0041472
.0031011
.279
.0053118
.0038363
0.250
0.0041835
0.0031245
0.280
0.0053598
0.0038620
.251
.0042199
.0031480
.281
.0054020
.0038877
.252
.0042566
.0031716
.282
.0054444
.0039135
.253
.0042934
.0031952
.283
.0054870
.0039394
.254
.0043305
.0032189
.284
.0055298
.0039654
0.255
0.0043677
0.0032427
0.285
0.0055728
0.0039914
.256
.0044051
.0032666
.286
.0056160
.0040175
.257
.0044427
.0032905
.287
.0056594
.0040437
.258
.0044804
.0033146
.288
.0057030
.0040700
.259
.0045184
.0033387
.289
.0057468
.0040963
0.260
0.0045566
0.0033628
0.290
0.0057908
0.0041227
.261
.0045949
.0033871
.291
.0058350
.0041491
.262
.0046334
.0034114
.292
.0058795
.0041757
.263
.0046721
.0034358
.293
.0059241
■ .0042023
.264
.0047111
.0034603
.294
.0059689
.0042290
0.265
0.0047502
0.0034848
0.295
0.0060139
0.0042557
.266
.0047894
.0035094
.296
.0060591
.0042826
.267
.0048289
.0035341
.297
.0061045
.0043095
.268
.0048686
.0035589
.298
.0061502
.0043364
.269
.0049085
.0035838
.299
.0061960
.0043635
.270
.0049485
.0036087
.300
.0062421
.0043906
TABLE la.
21
ELLIPSE.
HYPERBOLA. 1
A
LogE„
Log diff.
LogE,
Log diff.
LogE^
Log diff.
Log E,.
Log diff
0.000
0.0000000
9.2401
0.0000000
9.6378
0.0000000
9.2398
0.0000000
9.6378
.001
.0001738
.2403
9.9995656
.6381
9.9998263
.2395
.0004341
.6375
.002
.0003477
.2406
.9991309
.6384
.9996528
.2392
.0008680
.6372
.003
.0005217
.2408
.9986959
.6386
.9994794
.2389
.0013017
.6370
.004
.0006958
.2413
.9982607
.6389
.9993061
.2386
.0017350
.6367
0.005
0.0008701
9.2416
9.9978252
9.6391
9.9991329
9.2383
0.0021682
. 9.6365
.006
.0010445
.2418
.9973895
.6394
.9989598
.2381
.0026010
.6362
.007
.0012190
.2420
.9969535
.6396
.9987869
.2378
.0030337
.6360
.008
.0013936
.2423
.9965173
.6399
.9986141
.2375
.0034660
.6357
.009
.0015683
.2428
.9960807
.6402
.9984414
.2372
.0038981
.6354
0.010
0.0017432
9.2430
9.9956439
9.6405
9.9982688
9.2369
0.0043299
9.6352
.011
.0019182
.2433
.9952068
.6407
.9980963
.2366
.0047615
.6349
.012
.0020933
.2435
.9947695
.6410
.9979240
.2363
.0051928
.6347
.013
.0022685
.2438
.9943319
.6412
.9977517
.2360
.0056239
.6344
.014
.0024438
.2443
.9938941
.6414
.9975796
.2357
.0060547
.6342
0.015
0.0026193
9.2445
9.9934560
9.6417
9.9974076
9.2354
0.0064853
9.6339
.016
.0027949
.2448
.9930176
.6420
.9972357
.2351
.0069156
.6336
.017
.0029706
.2453
.9925789
.6423
.9970639
.2348
.0073456
.6334
.018
.0031465
.2455
.9921400
.6425
.9968923
.2345
.0077754
.6331
.019
.0033225
.2458
.9917008
.6428
.9967207
.2342
.0082049
.6329
0.020
0.0034986
9.2460
9.9912614
9.6430
9.99'65493
9.2339
0.0086342
9.6326
.021
.0036748
.2460
.9908217
.6433
.9963780
.2336
.0090632
.6323
.022
.0038510
.2465
.9903817
.6436
.9962068
.2333
.0094920
.6321
.023
.0040274
.2470
.9899415
.6438
.9960357
.2330
.0099205
.6318
.024
.0042040
.2472
.9895010
.6441
.9958648
.2328
.0103487
.6316
0.025
0.0043807
9.2475
9.9890602
9.6444
9.9956939
9.2325
0.0107767
9.6313
.026
.0045575
.2477
.9886192
.6446
.9955232
.2322
.0112045
.6311
.027
.0047344
.2480
.9881779
.6449
.9953526
.2319
.0116320
.6308
.028
.0049114
.2485
.9877363
.6452
.9951821
.2316
.0120592
.6306
.029
.0050886
.2487
.9872945
.6454
.9950117
.2313
.0124862
.6303 1
1
0.030
0.0052659
9.2490
9.9868524
9.6457
9.9948414
9.2310
0.0129130
9.6301 !
.031
.0054433
.2494
.9864100
.6459
.9946712
.2307
.0133395
.6298 i
.032
.0056209
.2497
.9859674
.6462
.9945012
.2304
.0137657
.6295
.033
.0057986
.2499
.9855245
.6465
.9943313
.2301
.0141917
.6293
.034
.0059764
.2502
.9850813
.6468
.9941615
.2298
.0146175
.6290
0.035
0.0061543
9.2504
9.9846378
9.6471
9.9939918
9.2295
0.0150430
9.6288
.036
.0063323
.2509
.9841940
.6474
.9938222
.2292
.0154683
.6285 1
.037
.0065105
.2512
.9837499
.6476
.9936528
.2290
.0158933
.6283 1
.038
.0066888
.2514
.9833056
.6478
.9934834
.2287
.0163180
.6280 1
.039
.0068672
.2516
.9828610
.6481
.9933142
.2284
.0167426
.6278
.040
.0070457
.2519
.9824161
.6484
.9931450
.2281
.0171668
.6275
22
ABLE la.
ELLIPSE.
HYPERBOLA.
A
Log E„
Log diflf.
LogE,
Log diff.
LogE^
Log diflf.
Log E,.
Log DiflT.
0.040
0.0070457
9.2519
9.9824161
9.6484
9.9931450
9.2281
0.0171668
9.6275
.041
.0072243
.2524
.9819709
.6487
.9929760
.2278
.0175908
.6273
.042
.0074031
.2526
.9815255
.6489
.9928071
.2275
.0180146
.6270
.043
.0075820
.2531
.9810798
.6492
.9',l2(;;>83
.2272
.0184381
.6267
.044
.0077611
.2533
.9806339
.6494
.9924696
.2269
.0188614
.6265
0.045
0.0079403
9.2536
9.9801877
9.6497
9.9923010
9.2266
0.0192844
9.6262
.046
.0081196
.2538
.9797412
.6500
.9921325
.2263
.0197072
.6260
.047
.0082990
.2543
.9792944
.6502
.9919642
.2260
.0201297
.6257
.048
.0084786
.2546
.9788474
.6505
.9917960
.2258
.0205520
.6255
.049
.0086583
.2548
.9784001
.6508
.9916279
.2255
.0209740
.6252
0.050
0.0088381
9.2550
9.9779525
9.6511
9.9914599
9.2252
0.0213958
9.6250
.051
.0090180
.2555
.9775046
.6514
.9912920
.2249
.0218174
.6247
.052
.0091981
.2558
.9770564
.6516
.9911242
.2246
.0222387
.6245
.053
.0093783
.2560
.9766079
.6519
.9909565
.2243
.0226597
.6242
.054
.0095586
.2565
.9761592
.6521
.9907890
.2240
.0230805
.6240
0.055
0.0097391
9.2567
9.9757102
9.6524
9.990G215
9.2237
0.0235011
9.6237
.056
.0099197
.2570
.9752609
.6527
.9904542
.2235
.0239214
.6235 !
.057
.0101004
.2572
.9748113
.6529
.9902869
.2232
.0243415
.6232 1
.058
.0102812
.2577
.9743615
.6532
.91101198
.2229
.0247614
.6230
.059
.0104622
.2579
.9739114
.6535
.9)S',)9528
.2226
.0251810
.6227 ;
0.060
0.0106433
9.2582
9.9734611
9.6538
9.9897859
9.2223
0.0256003
9.6225
.061
.0108245
.2584
.9730103
.6541
.9896191
.2220
.0260194
.6222
.062
.0110058
.2589
.9725593
.6543
.9894525
.2217
.0264383
.6220 ;
.063
.0111873
.2591
.9721080
.6546
.9.S'.)2S5:)
.2214
.0268570
.6217
.064
.0113689
.2594
.9716565
.6548
.9891195
.2211
.0272753
.6215
0.065
0.0115506
9.2598
9.9712047
9.6551
9.9889531
9.2208
0.0276935
9.6212
.066
.0117325
.2601
.9707526
.6554
.9887 869
.2206
.0281114
.6210
.067
.0119145
.2603
.9703002
.6557
.9SSG208
.2203
.0285291
.6207
.068
.0120966
.2606
.9698475
.6560
.9884548
.2200
.0289465
.6205
.069
.0122788
.2610
.9693945
.6562
.9882889
.2197
.0293637
.6202
0.070
0.0124612
9.2613
9.9689413
9.6565
9.9881231
9.2194
0.0297807
9.6200
.071
.0126437
.2617
.9684878
.6567
.9879574
.2191
.0301974
.6197
.072
.0128264
.2620
.9680340
.6570
.9877918
.2189
.0306139
.6195
.073
.0130092
.2622
.9675799
.6573
.9876263
.2186
.0310301.
.6192
.074
.0131921
.2625
.9671255
.6576
.9874610
.2183
.0314461
.6190
0.075
0.0133751
9.2629
9.9666708
9.6578
9.9872957
9.2180
0.0318618
9.6187
.076
.0135583
.2632
.9662159
.6581
.9871306
.2177
.0322773
.6185
.077
.0137416
.2634
.9657606
.6584
.9869655
.2174
.0326926
.6182
.078
.0139250
.2638
.9653051
.6587
.9868006
.2172
.0331076
.6180 '
.079
.0141086
.2641
.9648492
.6590
.9866358
.2169
.0335224
.6177
.080
.0142923
.2643
.9643931
.6592
.9864711
.2166
.0339370
.6175
i
TABLE la.
23
ELLIPSE.
HYPERBOLA.
A
LogE„
Log diff.
LogE^
Log diff.
LogE,
Log diff.
Log E,.
Log Diff.
0.080
0.0142923
9.2643
9.9643931
9.6592
9.9864711
9.2166
0.0339370
9.6175 '
.081
.0144761
.2646
.9639367
.6595
.9863065
.2163
.0343513
.6172
.082
.0146601
.2649
.9634800
.6598
.9861420
.2160
.0347654
.6170
.083
.0148442
.2652
.9630230
.6600
.9859776
.2157
.0351793
.6167
.084
.0150284
.2655
.9625657
.6603
.9858133
.2155
.0355930
.6165
0.085
0.0152128
9.2659
9.9621081
9.6606
9.9856491
9.2152
0.0360064
9.6163
.086
.0153973
.2662
.9616503
.6609
.9854850
.2149
.0364196
.6160
.087
.0155819
.2665
.9611922
.6611
.9853210
.2146
.0368325
.6158
.088
.0157667
.2668
.9607337
.6614
.9851572
.2143
.0372452
.6155
.089
.0159516
.2671
.9602749
.6617
.9849934
.2140
.0376577
.6153
0.090
0.0161367
9.2674
9.9598159
9.6620
9.9848298
9.2138
0.0380699
9.6150
.091
.0163218
.2677
.9593566
.6623
.9846663
.2135
.0384819
.6148
.092
.0165071
.2680
.9588970
.6625
.9845028
.2132
.0388937
.6145
.093
.0166925
.2684
.9584371
.6628
.9843395
.2129
.0393052
.6143
.094
.0168781
.2687
.9579769
.6631
.9841763
.2126
.0397165
.6141
0.095
0.0170638
9.2690
9.9575164
9.6634
9.9840132
9.2123
0.0401276
9.6138
■ .096
.0172497
.2693
.9570556
.6636
.9838502
.2121
.0405385
.6136
.097
.0174357
.2696
.9565945
.6639
.9836873
.2118
.0409491
.6133
.098
.0176218
.2700
.9561331
.6642
.9835245
.2115
.0413595
.6131
.099
.0178081
.2703
.9556714
.6645
.9833618
.2112
.0417696
.6128
0.100
0.0179945
9.2706
9.9552095
9.6648
9.9831992
9.2109
0.0421796
9.6126
.101
.0181810
.2708
.9547472
.6650
.9830367
.2107
.0425893
.6123
.102
.0183677
.2712
.9542847
.6653
.9828743
.2104
.0429988
.6121
.103
.0185545
.2715
.9538218
.6656
.9827121
.2101
.0434080
.6118
.104
.0187414
.2718
.9533586
.6659
.9825499
.2098
.0438170
.6116
0.105
0.0189285
9.2722
9.9528951
9.6662
9.9823879
9.2095
0.0442258
9.6114
.106
.0191157
.2725
.9524314
.6664
.9822259
.2093
.0446343
.6111
.107
.0193030
.2728
.9519673
.6666
.9820641
.2090
.0450426
.6109
.108
.0194905
.2731
.9515030
.6670
.9819023
.2087
.0454507
.6'106
.109
.0196781
.2734
.9510383
.6673
.9817407
.2084
.0458585
.6104
0.110
0.0198659
9.2738
9.9505734
9.6676
9.9815791
9.2081
0.0462661
9.6101
.111
.0200538
.2741
.9501081
.6678
.9814177
.2079
.0466735
.6099
.112
.0202418
.2744
.9496425
.6681
.9812563
.2076
.0470807
.6096
.113
.0204300
.2747
.9491766
.6684
.9810951
.2073
.0474876
.6094
.114
.0206183
.2750
.9487105
.6687
.9809340
.2070
.0478943
.6092
0.115
0.0208067
9.2754
9.9482440
9.6690
9.9807730
9.2067
0.0483008
9.6089
.116
.0209953
.2757
.9477772
.6692
.9806121
.2065
.0487071
.6087
.117
.0211840
.2760
.9473101
.6695
.9804513
.2062
.0491131
.6084
.118
.0213729
.2763
.9468428
.6698
.9802905
.2059
.0495189
.6082
.119
.0215619
.2767
.9463751
.6701
.9801299
.2056
.0499245
.6080
.120
.0217511
.2770
.9459071
.6704
.9799694
.2054
.0503298
.6077
24
TABLE Ic
ELLIPSE.
HYPERBOLA.
A
LogE„
Log diflf.
Log E^
Log diff.
Log E^
Log diff.
Log E,.
Log Diff.
0.120
0.0217511
9.2770
9.9459071
9.6704
9.9799694
9.2054
0.0503298
9.6077
.121
.0219404
.2773
.9454388
.6707
.9798090
.2051
.0507349
.6075
.122
.0221298
.2776
.9449702
.6709
.'.t7'.t(M87
.2048
.0511399
.6072
.123
.0223193
.2779
.9445013
.6712
.97'.) I.S85
.2045
.0515446
.6070
.124
.0225091
.2783
.9440321
.6715
.9793284
.2043
.0519490
.6068
0.125
0.0226990
9.2786
9.9435626
9.6718
9.9791684
9.2040
0.0523533
5.6065
.126
.0228889
.2789
.9430927
.6721
.9790085
.2037
.0527573
.6063
.127
.0230791
.2792
.9426226
.6724
.9788487
.2034
.0531611
.6061
.128
.0232693
.2795
.9421521
.6727
.9786890
.2032
.0535647
.6058
.129
.0234597
.2799
.9416813
.6729
.9785294
.2029
.0539681
.6056 :
0.130
0.0236503
9.2802
9.9412103
9.6732
9.9783699
9.2026
0.0543712
).6053
.131
.0238410
.2805
.9407389
.6735
.9782105
.2023
.0547741
.6051
.132
.0240318
.2808
.9402672
.6738
.9780512
.2021
.0551768
.6049
.133
.0242228
.2812
.9397952
.6741
.9778920
.2018
.0555793
.6046
.134
.0244139
.2815
.9393229
.6744
.9777329
.2015
.0559816
.6044
0.135
0.0246052
9.2818
9.9388503
9.6747
9.9775739
9.2012
0.0563836 <
).6041
.136
.0247966
.2822
.9383773
.6749
.9774150
.2010
.0567854
.6039
.137
.0249882
.2825
.9379041
.6752
.9772562
.2007
.0571870
.6037
.138
.0251799
.2828
.9374305
.6755
.9770975
.2004
.0575884
.6034
.139
.0253717
.2831
.9369567
.6758
.9769390
.2001
.0579895
.6032
0.140
0.0255637
9.2834
9.9364824
9.6761
9.9767805
9.1998
0.0583904 f
).6029
.141
.0257558
.2838
.9360079
.6764
.9766221
.1996
.0587911
.6027
.142
.0259481
.2841
.9355331
.6767
.9764638
.1993
.0591916
.6025
.143
.0261405
.2844
.9350580
.6770
.9763057
.1990
.0595919
.6022
.144
.0263331
.2848
.9345825
.6773
.9761476
.1988
.0599919
.6020
0.145
0.0265258
9.2851
9.9341067
9.6775
9.9759896
9.1985
0.0603917 {
).6018
.146
.0267187
.2854
.9336307
.6778
.9758317
.1982
.0607913
.6015
.147
.0269117
.2857
.9331543
.6781
.9756739
.1979
.0611907
.6013
.148
.0271048
.2861
.9326775
.6784
.9755162
.1977
.0615899
.6010
.149
.0272981
.2864
.9322005
.6787
.9753586
.1974
.0619888
.6008
0.150
0.0274915
9.2867
9.9317231
9.6790
9.9752011
9.1971
0.0623876 i
.6006
.151
.0276851
.2871
.9312455
.6793
.9750437
.1969
.0627861
.6003
.152
.0278789
.2874
.9307675
.6796
.9748864
.1966
.0631844
.6001
.153
.0280728
.2877
.9302892
.6798
.9747292
.1963
.0635825
.5999
. .154
.0282668
.2880
.9298106
.6801
.9745721
.1960
.0639804
.5996
0.155
0.0284610
9.2884
9.9293317
9.6804
9.9744151
9.1958
0.0643780 i
.5994
.156
.0286553
.2887
.9288524
.6807
.9742582
.1955
.0647755
.5992
.157
.0288498
.2890
.9283728
.6810
.9741014
.1952
.0651727
.5989
.158
.0290444
.2893
.9278929
.6813
.9739447
.1949
.0655697
.5987
.159
.0292392
.2897
.9274127
.6816
.9737881
.1946
.0659665
.5985
.160
.0294341
.2900
.9269321
.6819
.9736316
.1944
.0663631
.5982
TABLE la.
25
ELLIPSE.
HYPERBOLA.
A
LosE^
Log diflf.
LogE,
Log difF.
LogE^
Log diff.
LogE..
Log diff.
0.160
0.0294341
9.2900
9.9269321
9.6819
9.9736316
9.1944
0.0663631
9.5982
.161
.0296292
.2903
.9264512
.6822
.9734752
.1941
.0667595
.5980
.162
.0298243
.2906
.9259700
.6825
.9733189
.1938
.0671556
.5978
.163
.0300197
.2910
.9254885
.6828
.9731627
.1936
.0675516
.5975 i
.164
.0302152
.2913
.9250067
.6831
.9730066
.1933
.0679473
.5973
0.165
0.0304109
9.2916
9.9245245
9.6833
9.9728506
9.1930
0.0683428
9.5971
.166
.0306067
.2920
.9240421
.6836
.9726947
.1928
.0687381
.5968
.167
.0308026
.2923
.9235592
.6839
.9725389
.1925
.0691332
.5966
.168
.0309987
.2926
.9230761
.6842
.9723831
.1922
.0695281
.5963
.169
.0311949
.2930
.9225926
.6845
.9722275
.1920
.0699228
.5961
0.170
0.0313913
9.2933
9.9221089
9.6848
9.9720719
9.1917
0.0703172
9.5959
.171
.0315879
.2936
.9216247
.6851
.9719165
.1914
.0707114
.5956
.172
.0317846
.2940
.9211403
.6854
.9717611
.1912
.0711055
.5954
.173
.0319815
.2943
.9206555
.6857
.9716059
.1909
.0714993
.5952
.174
.0321784
.2946
.9201704
.6860
.9714507
.1906
.0718929
.5949
0.175
0.0323756
9.2950
9.9196850
9.6863
9.9712957
9.1904
0.0722863
9.5947
.176
.0325729
.2953
.9191992
.6866
.9711407
.1901
.0726795
.5945
.177
.0327704
.2956
.9187131
.6869
.9709859
.1898
.0730724
.5942
.178
.0329680
.2960
.9182266
.6872
.9708311
.1895
.0734652
.5940
.179
.0331657
.2963
.9177399
.6875
.9706764
.1893
.0738578
.5938
0.180
0.0333636
9.2966
9.9172528
9.6878
9.9705218
9.1890
0.0742501
9.5935
.181
.0335617
.2970
.9167654
.6881
.9703673
.1887
.0746422
.5933
.182
.0337599
.2973
.9162776
.6884
.9702129
.1885
.0750341
.5931
.183
.0339582
.2977
.9157895
.6886
.9700587
.1882
.0754259
.5928
.184
.0341568
.2980
.9153011
.6889
.9699045
.1879
.0758173
.5926
0.185
0.0343555
9.2983
9.9148123
9.6892
9.9697504
9.1877
0.0762086
9.5924
.186
.0345543
.2987
.9143232
.6895
.9695964
.1874
.0765997
.5922
.187
.0347533
.2990
.9138338
.6898
.9694425
.1871
.0769906
.5919
.188
.0349524
.2993
.9133441
.6901
.9692887
.1869
.0773812
.5917
.189
.0351517
.2997
.9128540
.6904
.9691350
.1866
.0777717
.5915
0.190
0.0353511
9.3000
9.9123635
9.6907
9.9689813
9.1863
0.0781619
9.5912
.191
.0355507
.3003
.9118727
.6910
.9688278
.1861
.0785520
.5910
.192
.0357505
.3007
.9113816
.6913
.9686743
.1858
.0789418
.5908
.193
.0359504
.3010
.9108901
.6916
.9685210
.1855
.0793315
.5906
.194
.0361505
.3014
.9103983
.6919
.9683678
.1853
.0797209
.5903
0.195
0.0363507
9.3017
9.9099062
9.6922
9.9682146
9.1850
0.0801102
9.5901
.196
.0365511
.3-020
.9094138
.6925
.9680615
.1847
.0804992
.5899
.197
.0367516
.3024
.9089210
.6928
.9679086
.1845
.0808881
.5896
.198
.0369523
.3027
.9084278
.6931
.9677557
.1842
.0812767
.5894
.199
.0371532
.3031
.9079343
.6934
.9676029
.1839
.0816651
.5892
.200
.0373542
.3034
.9074405
.6937
.9674502
.1837
.0820533
.5889
26
TABLE la.
ELLIPSE.
HYPERBOLA.
A
Lo- E,,.
0.0373542
Log diff.
Log E,
Log difl-.
Log E,.
Log diff.
Log E,.
Log DiflF.
0.200
9.3034
9.9074405
9.6937
9.9674502
9.1837
0.0820533
9.5889
.201
.0375554
.3037
.9069463
.6940
.9672976
.1834
.0824413
.5887
.202
.0377567
.3041
.9064518
.6943
.9671451
.1831
.0828291
.5885
.203
.0379582
.3044
.9059569
.6946
.9669927
.1829
.0832166
.5882
.204
.0381598
.3047
.9054617
.6949
.9668404
.1826
.0836040
.5880
0.205
0.0383616
9.3051
9.9049662
9.6952
9.9666882
9.1823
0.0839911
9.5878
.206
.0385635
.3054
.9044703
.6955
.9665361
.1821
.0843781
.5876
.207
.0387656
.3058
.9039741
.6958
.9G(;;->84i
.1818
.0847649
.5873
.208
.0389679
.3061
.9034775
.6961
.:)G62321
.1815
.0851514
.5871
.209
.0391703
.3065
.9029806
.6964
.9660803
.1813
.0855377
.5869
0.210
0.0393729
9.3068
9.9024833
9.6967
9.9659285
9.1810
0.0859239
9.5867
.211
.0395757
.3071
.9019857
.6970
.9657768
.1808
.0863099
.5864
.212
.0397786
.3075
.9014877
.6974
.9656253
.1805
.0866956
.5862
.213
.0399817
.3078
.9009894
.6977
.9654738
.1802
.0870812
.5860
.214
.0401849
.3081
.9004907
.6980
.9653224
.1800
.0874665
.5858
0.215
0.0403883
9.3085
9.8999917
9.6983
9.9651711
9.1797
0.0878517
9.5855
.216
.0405918
.3088
.8994924
.6986
.9650199
.1795
.0882367
.5853
.217
.0407955
.3092
.8989927
.6989
.9648687
.1792
.0886214
.5851
.218
.0409994
.3095
.8984927
.6992
.9647177
.1789
.0890060
.5849
.219
.0412034
.3099
.8979923
.6995
.9645667
.1787
.0893903
.5846
0.220
0.0414076
9.3102
9.8974915
9.6998
9.9644159
9.1784
0.0897745
9.5844
.221
.0416120
.3106
.8969904
.7001
.9642651
.1782
.0901585
.5842
.222
.0418165
.3109
.8964889
.7004
.9641145
.1779
.0905422
.5839
.223
.0420211
.3112
.8959881
.7007
.9639639
.1776
.0909258
.5837
.224
.0422260
.3116
.8954849
.7010
.9638134
.1774
.0913091
.5835
0.225
0.0424310
9.3119
9.8949824
9.7013
9.9636630
9.1771
0.0916923
9.5833
.226
.0426362
.3123
.8944795
.7016
.9635127
.1768
.0920753
.5830
.227
.0428415
.3127
.8939762
.7019
.9633625
.1766
.0924580
.5828
.228
.0430470
.3130
.8934726
.7022
.9632123
.1763
.0928405
.5826
.229
.0432527
.3133
.8929687
.7025
.9630623
.1760
.0932229
.5823
0.230
0.0434585
9.3137
9.8924644
9.7028
9.9629124
9.1758
0.0936050
9.5821
.231
.0436645
.3140
.8919597
.7031
.9627625
.1755
.0939870
.5819
.232
.0438707
.3144
.8914547
.7035
.9626128
.1752
.0943687
.5817
.233
.0440770
.3147
.8909493
.7038
.9624631
.1750
.0947503
.5814
.234
.0442835
.3151
.8904436
.7041
.9623136
.1747
.0951317
.5812
0.235
0.0444902
9.3154
9.8899375
9.7044
9.9621641
9.1745
0.0955128
9.5810
.236
.0440970
.3158
.8894310
.7047
.9620147
.1742
.0958938
.5808
.237
.0419040
.3161
.8889242
.7050
.9618654
.1740
.0962745
.5806
.238
.0451111
.3165
.8884170
.7053
.9617162
.1737
.0966551
.5803
.239
.0453184
.3168
.8879094
.7056
.9615670
.1734
.0970355
.5801
.240
.0455259
.3171
.8874015
.7059
.9614180
.1732
.0974157
.5799
TABLE Ic
27
ELLIPSE.
HYPERBOLA.
A
l^og E„
Log diif.
LogE,
Log difF.
LogE„
Log diff.
LogE,.
Log Diff.
0.240
0.0455259
9.3171
9.8874015
9.7059
9.9614180
9.1732
0.0974157
9.5799
.241
.0457335
.3175
.8868932
.7063
.9612690
.1729
.0977957
.5797
.242
.0459413
.3179
.8863846
.7066
.9611202
.1727
.0981755
.5794
.243
.0461493
.3182
.8858756
.7069
.9609714
.1724
.0985551
.5792
.244
.0463575
.3186
.8853663
.7072
.9608227
.1722
.0989345
.5790
0.245
0.0465658
9.3189
9.8848566
9.7075
9.9606741
9.1719
0.0993137
9.5788
.246
.0467743
.3193
.8843465
.7078
.9605256
.1716
.0996927
.5786
.247
.0469830
.3196
.8838360
.7081
.9603771
.1714
.1000716
.5783
.248
.0471918
.3200
.8833252
.7084
.9602288
.1711
.1004502
.5781
.249
.0474008
.3203
.8828140
.7087
.9600805
.1709
.1008287
.5779
0.250
0.0476099
9.3207
9.8823025
9.7090
9.9599324
9.1706
0.1012069
9.5777
.251
.0478193
.3210
.8817906
.7094
.9597843
.1704
.1015850
.5775
.252
.0480288
.3214
.8812783
.7097
.9596363
.1701
.1019628
.5772
.253
.0482385
• .3217
.8807657
.7100
.9594884
.1698
.1023405
.5770
.254
.0484483
.3221
.8802526
.7103
.9593406
.1696
.1027180
.5768
0.255
0.0486583
9.3224
9.8797392
9.7106
9.9591929
9.1693
0.1030953
9.5766
.256
.0488685
.3226
.8792254
.7109
.9590453
.1691
.1034724
.5763
.257
.0490788
.3231
.8787113
.7112
.9588977
.1688
.1038493
.5761
.258
.0492893
.3235
.8781968
.7116
.9587502
.1685
.1042259
.5759
.259
.0495000
.3238
.8776819
.7119
.9586029
.1683
.1046024
.5756
0.260
0.0497109
9.3242
9.8771666
9.7122
9.9584556
9.1680
0.1049787
9.5754
.261
.0499219
.3245
.8766510
.7125
.9583084
.1678
.1053548
.5752
.262
.0501331
.3249
.8761350
.7128
.9581613
.1675
.1057308
.5750
.263
.0503445
.3252
.8756186
.7131
.9580143
.1673
.1061065
.5748
.264
.0505560
.3256
.8751019
.7134
.9578673
.1670
.1064821
.5746
0.265
0.0507677
9.3260
9.8745848
9.7137
9.9577205
9.1668
0.1068574
9.5743
.266
.0509796
.3263
.8740673
.7141
.9575737
.1665
.1072326
.5741
.267
.0511917
.3267
.8735495
.7144
.9574270
.1662
.1076076
.5739
.268
.0514040
.3270
.8730312
.7147
.9572804
.1660
.1079824
.5737
.269
.0516164
.3274
'.8725126
.7150
.9571339
.1657
.1083570
.5735
0.270
0.0518290
9.3277
9.8719936
9.7153
9.9569875
9.1655
0.1087314
9.5733
.271
.0520418
.3281
.8714742
.7157
.9568412
.1652
.1091056
.5730
.272
.0522547
.3284
.8709544
.7160
.9566949
.1650
.1094797
.5728
.273
.0524678
.3288
.8704343
.7163
.9565487
.1647
.1098536
.5726
.274
.0526811
.3292
.8699137
.7166
.9564027
.1644
.1102272
.5724
0.275
0.0528946
9.3295
9.8693928
9.7169
9.9562567
9.1642
0.1106007
9.5722
.276
.0531082
.3299
.8688715
.7173
.9561108
.1639
.1109740
.5719
.277
.0533220
.3303
.8683498
.7176
.9559650
.1637
.1113471
.5717
.278
.0535360
.3306
.8678278
.7179
.9558193
.1634
.1117200
.5715
.279
.0537502
.3310
.8673053
.7182
.9556736
.1632
.1120927
.5713
.280
.0539646
.3313
.8667825
.7185
.9555281
.1629
.1124652
.5710
28
TABLE la.
ELLIPSE.
HYPERBOLA.
A
LogE„
Log diff.
Log E,
Log diff.
Log E^
Log diff.
LogE,..
Log Diff.
0.280
0.0539646
9.3313
9.8667825
9.7185
9.9555281
9.1629
0.1124652
9.5710
.281
.0541791
.3317
.8662593
.7188
.9553826
.1627
.1128375
.5708
.282
.0543939
.3320
.8657357
.7192
.9552372
.1624
.1132097
.5707
.283
.0546087
.3324
.8652117
.7195
.9550919
.1622
.1135817
.5704
.284
.0548238
.3327
.8646873
.7198
.9549467
.1619
.1139534
.5701
0.285
0.0550390
9.3331
9.8641625
9.7201
9.9548015
9.1617
0.1143250
9.5699
.286
.0552546
• .3335
.8636374
.7204
.9546564
.1614
.1146964
.5698
.287
.0554700
.3338
.8631118
.7208
.9545115
.1612
.1150677
.5695
.288
.0556858
.3342
.8625859
.7211
.9543666
.1609
.1154387
.5693
.289
.0559018
.3345
.8620596
.7214
.'9542218
.1606
.1158096
.5691
0.290
0.0561179
9.3349
9.8615329
9.7217
9.9540771
9.1604
0.1161803
9.5689
.291
.0563342
.3353
.8610058
.7221
.9539325
.1601
.1165508
.5687
.292
.0565507
.3356
.8604783
.7224
.9537879
.1599
.1169211
.5685
.293
.0567674
.3360
.8599504
.7227
.9536435
.1596
.1172913
.5683
.294
.0569842
.3364
.8594221
.7230
.9534991
.1594
.1176612
.5680
0.295
0.0572013
9.3367
9.8588935
9.7233
9.9533548
9.1591
0.1180310
9.5678
.296
.0574185
.3371
.8583644
.7236
.9532106
.1589
.1184006
.5675 1
.297
.0576359
.3375
.8578349
.7240
.9530665
.1586
.1187699
.5673
.298
.0578535
.3379
.8573051
.7243
.9529224
.1584
.1191391
.5671
.299
.0580713
.3383
.8567748
.7246
.9527785
.1581
.1195081
.5668
.300
.0582893
.3387
.8562442
.7249
.9526346
.1578
0.1198768
9.5666
TABLE Ua.
29
ro-
Vo-
LogAi.
LogAj.
Log A3.
0
6 0 0.00
4-3.7005216
—0.00000
—9.695
2
2 47 11.83
3.7000079
0.47160
9.691
4
5 34 0.00
3.6984710
0.76930
9.681
6
8 20 1.19
3.6959236
0.93987
9.664
8
11 4 52.82
3.6923863
1.05702
9.641
10
J 3 48 13.31
+3.6878872
—1.14430
—9.610
12
16 29 42.39
3.6824613
1.21171
9.571
14
19 9 1.36
3.6761493
1.26497
9.525
16
21 45 53.23
3.6689972
1.30744
9.470
18
24 20 2.89
3.6610547
1.34135
9.405
20
26 51 17.15
+3.6523748
—1.36825
—9.329
22
29 19 24.78
3.6430121
1.38929
9.239
24
31 44 16.52
3.6330224
1.40535
9.130
26
34 5 44.97
3.6224621
1.41714
8.994
28
36 23 44.51
3.6113863
1.42520
8.814
30
38 38 11.23
+3.5998496
—1.43003
—8.538
32
40 49 2.74
3.5879044
1.43201
—7.847
34
42 56 18.02
3.5756011
1.43149
+8.237
36
44 59 57.33
3.5629877
1.42877
8.585
38
47 0 2.00
3.5501091
1.42410
8.753
40
48 56 34.33
+3.5370077
—1.41772
+8.857
42
50 49 37.39
3.5237227
1.40983
8.928
44
52 39 14.95
3.5102905
1.40060
8.978
46
54 25 31.32
3.4967444
1.39U20
9.013
48
56 8 31.24
3.4831149
1.37878
9.038
50
57 48 19.82
+3.4694297
—1.36645
+9.056
52
59 25 2.41
3.4557140
1.35333
9.067
54
60 58 44.53
3.4419903
1.33952
9.073
56
62 29 31.82
3.4282790
1.32512
9.076
58
63 57 29.99
3.4145981
1.31021
9.075
60
65 22 44.74
+3.4009637
—1.29486
+9.071
64
68 5 26.60
3.3738900
1.26308
9.056
68
70 38 21.86
3.3471520
1.23025
9.035
72
73 2 13.17
3.3208214
1.19672
9.008
76
75 17 40.91
3.2949510
1.16277
8.978
80
77 25 22.94
+3.2695785
—1.12863
+8.945
84
79 25 54.44
3.2447291
1.09447
8.910
88
81 19 47.97
3.2204185
1.06044
8.874
92
83 7 33.52
3.1966546
1.02665
8.837
96
84 49 38.62
3.1734393
0.99319
8.798
100
86 26 28.52
+3.1507694
—0.96012
+8.760
104
87 58 26.32
3.1286388
0.92749
8.721
108
89 25 53.18
3.1070382
0.89534
8.682
112
90 49 8.43
3.0859565
0.86370
8.643
116
92 8 29.76
3.0653811
0.83257
8.605
30
TABLE Ila.
To.
Vo.
Log Ai .
LogAo.
Log A3.
116
9°2 8 29.76
+3.0653811
—0.83257
+8.605
120
93 24 13.33
3.0452984
0.80199
8.567
124
94 36 33.98
3.0256943
0.77194
8.529
128
95 45 45.25
3.0065544
0.74244
8.491
132
96 51 59.60
2.9878638
0.71347
8.454
136
97 55 28.43
+2.9696079
—0.68505
+8.418
140
98 56 22.24
2.9517723
0.65716
8.382
144
99 54 50.68
2.9343427
0.62979
8.346
148
100 51 2.62
2.9173052
0.60293
8.311
152
101 45 6.25
2.9006462
0.57658
8.276
156
102 37 9.12
+2.8843526
—0.55071
+8.242
160
103 27 18.23
2.8684116
0.52534
8.209
164
104 15 40.03
2.8528110
0.50043
8.176
168
J 05 2 20.49
2.8375388
0.47598
8.143
172
105 47 25.18
2.8225838
0.45198
8.111
176
106 30 59.23
+2.8079349
—0.42841
+8.080
180
107 13 7.45
2.7935817
0.40526
8.049
184
107 53 54.28
2.7795141
0.38253
8.018
188
108 33 23.87
2.7657223
0.36020
7.988
192
109 11 40.10
2.7521971
0.33826
7.959
196
109 48 46.58
+2.7389297
—0.31670
+7.9.30
200
110 24 46.69
2.7259114
0.29551
7.901
210
111 50 16.87
2.6944032
0.24407
7.831
220
113 9 55.67
2.6642838
0.19472
7.764
230
114 24 20.89
2.6354467
0.14732
7.700
240
115 34 4.97
+2.6077961
—0.10174
+7.637
250
116 39 35.94
2.5812455
0.05786
7.577
260
117 41 18.16
2.5557170
0.01556
7.519
270
118 39 32.86
2.5311401
9.97476
7.463
280
119 34 38.67
2.5074507
9.93535
7.409
290
120 26 51.98
+2.4845910
—9.89725
+7.356
300
121 16 27.30
2.4625078
9.86U38
7.305
310
122 3 37.49
2.4411532
9.82467
7.256
320
122 48 34.01
2.4204831
9.79006
7.208
330
123 31 27.11
2.4004569
9.75648
7.161
340
124 12 25.97
+2.3810379
-9.7-2387
+7.116
350
124 51 38.87
2.3621918
9.69219
7.072
360
125 29 13.25
2.3438873
9.66139
7.029
370
126 5 15.87
2.3260956
9.63142
6.987
380
126 39 52.85
2.3087898
9.60224
6.947
390
127 13 9.75
+2.2919450
—9.57381
+6.907
400
127 45 11.66
2.2755384
9.54610
6.868
420
128 45 48.63
2.2439555
9.49269
6.794
440
129 42 16.43
2.2138871
9.44176
6.723
460
130 35 2.66
2.1851991
9.39310
6.655
TABLE Ila.
31
1
j to.
Vo.
LogAi.
LogA2.
LogA^.
460
130 3o 2.66
+2.1851991
—9.39310
+96.655
480
131 24 30.82
2.1577741
9.34654
6.589
500
132 11 1.09
2.1315086
9.30188
6.527
1 520
132 54 50.84
2.1063114
9.25901
6.467
540
133 36 15.19
2.0821011
9.21777
6.409
560
134 15 27.33
+2.0588051
—9.17805
+96.353
580
134 52 38.80
2.0363588
9.13976
6.299
600
135 27 59.81
2.0147037
9.10278
6.247
640
136 33 45.52
1.9735615
9.03246
6.148
680
137 33 45.39
1.9350140
8.96649
6.055
720
138 28 48.27
+1.8987593
—8.90438
+95.968
760
139 19 33.81
1.8645446
8.84571
5.885
800
140 6 34.57
1.8321564
8.79012
5.807
850
141 0 45.22
1.7939648
8.72451
5.714
900
141 50 30.05
1.7580440
8.66275
5.627
950
142 36 24.37
+1.7241428
—8.60441
+95.544
1000
143 18 57.20
1.6920492
8.54915
5.466
1050
143 58 32.66
1.6615826
8.49.665
5.392
1100
144 35 30.95
1.6325881
8.44666
5.321
1150
145 10 9.20
1.6049315
8.39896
5.254
1200
145 42 41.98
+1.5784963
—8.35333
+95.189
1250
146 13 21.82
1.5531804
8.30962
5.127
1300
146 42 19.55
1.5288937
8.26767
5.068
1350
147 9 44.57
1.5055568
8.22735
5.011
1400
147 35 45.11
1.4830989
8.18853
4.956
1450
148 0 28.40
+1.4614567
—8.15110
+94.903
1500
148 24 0.83
1.4405738
8.11498
4.851
1600
149 7 55.10
1.4008865
8.04631
4.754
1700
149 48 6.25
1.3636849
7.98190
4.663
1800
150 25 5.10
1.3286785
7.92126
4.576
1900
150 59 16.75
+1.2956243
, —7.86398
+94.495
2000
151 31 1.89
1.2643177
7.80971
4.418
2100
152 0 37.76
1.2345845
7.75814
4.345
2200
152 28 18.85
1.2062750
7.70903
4.275
2300
152 54 17.45
1.1792601
7.66216
4.208
2400
153 18 44.05
+1.1534272
—7.61732
+94.145
2500
153 41 47.70
1.1286779
7.57435
4.084
2600
154 3 36.21
1.1049254
7.53310
4.025
2700
154 24 16.39
1.0820930
7.49344
3.969
2800
154 43 54.21
1.0601125
7.45526
3.914
2900
155 2 34.93
+1.0389230
—7.41844
+93.862
3000
155 20 23.19
1.0184698
7.38289
3.811
3200
155 53 38.39
0.9795803
7.31529
3.715
3400
156 24 7.80
0.9431040
7.25186
3.625
3600
156 52 14.00
0.9087603
7.19213
3.540
32
TABLE Ila.
To.
fo-
Log Ai .
LogA2.
Log As-
3600
15°6 52 14.00
4-0.9087603
—97.19213
+93.540
3800
157 18 15.42
0.8763145
7.13568
3.459
4000
157 42 27.29
0.8455688
7.08218
3.383
4200
158 5 2.33
0.8163545
7.03133
3.311
4400
158 26 11.25
0.7885269
6.98289
3.242
4600
158 46 3.15
-^0.7619607
—96.93664
+93.176
4800
159 4 45.83
0.7365469
6.89238
3.113
5000
159 22 25.99
0.7121902
6.84996
3.053
5200
159 39 9.45
0.6888063
6.80923
2.995
5600
160 10 6.00
0.6446674
6.73234
2.885
6000
160 38 9.17
+0.6036264
—96.66082
+92.783
6400
161 3 45.36
0.5652780
6.59398
2.688
6800
161 27 15.57
0.5292915
6.53125
2.599
7200
J 61 48 56.78
0.4953934
6.47215
2.514
7600
162 9 2.89
0.4633554
6.41629
2.435
8000
162 27 45.39
+0.4329843
—96.36332
f92.359
8400
162 45 13.90
0.4041157
6.31297
2.287
8800
163 1 36.52
0.3766081
6.26499
2.219
9200
163 17 0.16
0.3503393
6.21916
2.154
9600
163 31 30.72
0.3252029
6.17531
2.091
10000
163 45 13.32
+0.3011054
—96.13326
+92.031
10500
164 1 20.80
0.2723199
6.08303
1.959
11000
164 16 27.66
0.2448894
6.03516
1.891
11500
164 30 40.23
0.2186921
5.98944
1.826
12000
164 44 3.94
0.1936223
5.94568
1.764
13000
165 8 42.90
+0.1465042
—95.86343
+91.646
14000
165 30 55.26
0.1029147
5.78733
1.538
15000
165 51 4.63
0.0623627
5.71652
1.437
16000
166 9 29.58
0.0244528
5.65032
1.342
17000
166 26 24.88
9.9888624
5.58817
1.254
18000
166 42 2.53
+9.9553241
—95.52959
+91.170
19200
166 59 18.90
9.9174751
5.46348
1.076
20400
167 15 11.32
9.8819393
5.40141
90.987
21600
167 29 51.00
9.8484507
5.34290
90.904
22800
167 43 27.11
9.8167866
5.28758
90.825
24000
167 56 7.28
+9.7867585
—95.23512
+90.750
26000
168 15 26.77
9.7399215
5.15328
90.633
28000
168 32 51.95
9.6965794
5.07755
90.525
30000
168 48 41.17
9.6562474
5.00706
90.424
32000
169 3 8.84
9.6185347
4.94116
90.330
34000
169 16 26.46
+9.5831221
—94.87926
+90.242
36000
169 28 43.36
9.5497452
4.82093
90.159
38000
169 40 7.19
9.5181828
4.76576
90.080
40000
169 50 44.28
9.4882481
4.71343
90.005
TABLE Ilia.
33
■n
Log ^.
Log Diff.
V
Log 11.
Log Diff.
V
Log fi.
Log Diff.
0.00
.01
.02
0.00000 00
.00000 18
.00000 72
1.556
1.857
0.30
.31
.32
0.00167 33
.00179 01
.00191 12
3.0594
.0754
.0910
0.60
.61
.62
0.00735 26
.00763 61
.00792 74
3.4468
.4585
.4703
0.03
.04
.05
0.00001 62
.00002 89
.00004 52
2.0354
.1614
.2589
0.33
.34
.35
0.00203 67
.00216 66
.00230 10
3 1062
.1211
.1356
0.63
.64
.65
0.00822 68
.00853 45
.00885 08
3.4822
.4941
.5061
0.06
.07
.08
.00006 52
.00008 88
.00011 61
2.3385
.4057
.4639
0.36
.37
.38
0.00243 99
.00258 34
.00273 15
3.1498
.1638
.1774
0.66
.67
.68
0.00917 59
.00951 03
.00985 42
3.5182
.5304
5427
1 0.09
i .10
.11
0.00014 70
.00018 16
.00021 99
2.5152
.5617
.6031
0.39
.40
.41
0.00288 43
.00304 20
.00320 45
3.1911
.2044
.2175
0.69
.70
.71
0.01020 81
.01057 23
.01094 73
3.5551
.5677
.5805
0.12
.13
.14
0.00026 18
.00030 74
.00035 68
2.6410
.6767
.7097
0.42
.43
.44
0.00337 20
.00354 45
.00372 22
3.2304
.2433
.2557
0.72
.73
.74
0.01133 35
.01173 15
.01214 19
3.5934
.6066
.6200
0.15
.16
.17
0.00040 99
.00046 68
.00052 75
2.7404
.7694
.7966
0.45
.46
.47
0.00390 50
.00409 31
.00428 67
3.2681
.2807
.2930
0.75
.76
.77
0.01256 52
.01300 22
.01345 36
3.6336
.6476
.6618
0.18
.19
.20
0.00059 20
.00066 03
.00073 25
2.8222
.8466
.8701
0.48
.49
.50
0.00448 58
.00469 06
.00490 11
3.3053
•3173
.3293
0.78
.79
.80
0.01392 02
.01440 31
.01490 32
3.6765
.6915
.7070
0.21
.22
.23
0.00080 86
.00088 86
.00097 25
2.8924
.9135
.9340
0.51
.52
.53
0.00511 75
.00533 98
.00556 83
3.3411
.3529
.3647
0.81
.82
.83
0.01542 18
.01596 03
.01652 02
3.7231
.7397
.7570
0.24
.25
.26
0.00106 04
.00115 23
.00124 83
2.9538
.9729
.9914
0.54
.55
.56
0.00580 30
.00604 41
.00629 19
3.3764
.3882
.4000
0.84
.85
.86
0.01710 33
.01771 19
.01834 86
3.7751
.7942
.8144
0.27
.28
.29
.00134 84
.00145 25
.00156 08
3.0090
.0261
.0430
0.57
.58
.59
0.00654 65
.00680 80
.00707 66
3.4117
.4233
.4350
0.87
.88
.89
0.01901 65
.01971 95
.02046 29
3.8360
.8593
.8846
0.30
.31
.32
0.00167 33
.00179 01
.00191 12
3.0594
.0754
.0910
0.60
.61
0.00735 26
.00763 61
.00792 74
3.4468
.4585
.4703
0.90
.91
.92
0.02125 29
.02209 92
.02301 60
3.9128
.9452
i
34
TABLE IV,
m sin z* = sin (z — y)
m and
q positive
'i
's
0'
^
^m
z""
q
^'
o
m"
m'
m'
vf
m"
m!
m
'
«/'
o
O 4
o
o
O 1
o
o /
^
^
1
4.2976
9.9999
1 0
1 20
1 20
89 40
89 40
177 37
180
5o
181 b
2
3.3950
9.9996
2 0
2 40
2 40
89 20
89 20
175 14
181
51
182 0
3
2.8675
9.9992
3 0
4 0
4 0
89 0
89 0
172 52
182
46
183 0
4
2.4938
9.9986
4 0
5 20
5 20
88 40
88 40
170 28
183
42
184 0
5
2.2044
9.9978
5 0
6 41
6 41
88 19
88 19
168 5
184 37
185 0
6
1.9686
9.9968
6 0
8 1
8 1
87 59
87 59
165 41
185
32
186 0
7
1.7698
9.9957
7 1
9 22
9 22
87 38
87 38
163 18
186
28
186 59
8
1.5981
9.9943
8 1
10 42
10 42
87 18
87 18
160 52
187
23
187 59
9
1.4473
9.9928
9 2
12 3
12 3
86 57
86 57
158 28
188
18
188 58
10
1.3130
9.9911
10 3
13 25
13 25
86 35
86 35
156 3
189
13
189 57
11
1.1922
9.9892
11 5
14 46
14 46
86 14
86 14
153 37
190
9
190 56
12
1.0824
9.9871
12 7
16 8
16 8
85 b-l
85 52
151 10
191
4
191 54
13
0.9821
9.9848
13 9
17 31
17 31
85 29
85 29
148 43
191
59
192 52
14
0.8898
9.9823
14 12
18 53
18 53
85 7
85 7
146 14
192
54
193 49
15
0.8045
9.9796
15 16
20 17
20 17
84 43
84 43
143 45
193
49
194 46
16
0.7254
9.9767
16 20
21 40
21 40
84 20
84 20
141 14
194
44.
195 42
17
0.6518
9.9736
17 26
23 5
23 5
83 55
83 55
138 42
195
39
196 38
18
0.5830
9.9702
18 33
24 30
24 30
83 30
83 30
136 9
196
33
197 33
19
0.5185
9.9667
19 41
25 56
25 56
83 4
83 4
133 34
197
28
198 28
20
0.4581
9.9629
20 51
27 23
27 23
82 37
82 37
130 58
198
23
199 22
21
0.4013
9.9588
22 2
28 50
28 50
82 10
82 10
128 19
199
17
200 15
22
0.3479
9.9545
23 15
30 19
30 19
81 41
81 41
125 38
200
11
201 8
23
0.2976
9.9499
24 31
31 49
31 49
81 11
81 11
122 55
201
6
202 0
24
0.2501
9.9451
25 49
33 20
33 20
80 40
80 40
120 9
202
0
202 51
25
0.2053
9.9400
27 10
34 53
34 53
80 7
80 7
117 20
202
54
203 42
26
0.1631
9.9345
28 35
36 28
36 28
79 32
79 32
114 27
203
47
204 32
27
0.1232
9.9287
30 4
38 5
38 5
78 55
78 55
111 30
204
41
205 22
28
0.0857
9.9226
31 38
39 45
39 45
78 15
78 15
108 27
205
35
206 11
29
0.0503
9.9161
33 18
41 27
41 27
77 33
77 33
105 19
206
28
207 0
30
0.0170
9.9092
35 5
43 13
43 13
76 47
76 47
102 3
207
21
207 48
31
9.9857
9.9019
37 1
45 4
45 4
75 56
75 56
98 37
208
14
208 36
32
9.9565
9.8940
39 9
47 1
47 1
74 59
74 59
95 0
209
6
209 24
33
9.0292
9.8856
41 33
49 6
49 6
73 54
73 54
91 6
209
58
210 11
1 34
9.9040
9.8765
44 21
51 22
51 22
72 38
72 38
86 49
210
50
210 58
1 35
9.8808
9.8665
47 47
53 58
53 58
71 2
71 2
81 53
211
41
211 46
36
9.8600
9.8555
52 31
57 13
57 13
68 47
68 47
75 40
212
32
212 33
9'
9.8443
9.8443
63 26
63 26
63 26
63 26
63 26
63 26
213
15
213 15
q' =
36° 52' 11.64"
sin c
f = OS.
TABLE IV
35
in sin z* =
=:sin(a: + 9).
m
and q positive.
'g
'^
2'
0°
^m
2
IT
9.
-i
fcc
m
'
m
/
m
'
m
m
/
m
'
m
m
^
^
o
0
o
o
^
^
^
1
4.2976
9.9999
2
23
90
20
90
20
178
40
178
40
179
0
359
0
359 0
2
3.3950
9.9996
4
46
90
40
90
40
177
20
177
20
178
0
358
0
358 9
3
2.8675
9.9992
7
8
91
0
91
0
175
0
175
0
177
0
357
0
357 14
4
2.4938
9.9986
9
32
91
20
91
20
174
40
174
40
176
0
356
0
356 18
5
2.2044
9.9978
11
55
91
41
91
41
173
19
173
19
175
0
355
0
355 23
6
1.9686
9.9968
14
19
92
1
92
1
171
59
171
59
174
0
354
0
354 28
7
1.7698
9.9957
16
42
92
22
92
22
170
38
170
38
172
59
353
1
353 32
8
1.5981
9.9943
19
7
92
42
92
42
169
18
169
18
171
59
352
1
352 37
9
1.4473
9.9928
21
32
93
3
93
3
167
57
167
57
170
58
351
2
351 42
10
1.3130
9.9911
23
57
93
25
93
25
166
35
166
35
169
57
350
3
350 47
11
1.1922
9.9892
26
23
93
46
93
46
165
14
165
14
168
55
349
4
349 51
12
1.0824
9.9871
28
50
94
8
94
8
163
52
163
52
167
54
348
6
348 56
13
0.9821
9.9848
31
17
94
31
94 31
162
29
162
29
166
51
347
,8
348 1
14
0.8898
9.9823
33
46
94
53
94
53
161
7
161
7
165
48
346
11
347 6
15
0.8045
9.9796
36
15
95
17
95
17
159
43
159
43
164
44
345
14
346 11
16
0.7254
9.9767
38
46
95
40
95
40
158
20
158
20
163
40
344
18
345 16
17
0.6518
9.9736
41
18
96
5
96
5
156
55
156
55
162
34
343
22
344 21
18
0.5830
9.9702
43
51
96
30
96
30
155
30
155
30
161
27
342
27
343 27
19
0.5185
9.9667
46
26
96
56
96
56
154
4
154
4
160
19
341
32
342 32
20
0.4581
9.9629
49
2
97
23
97
23
1.52
37
152
37
159
9
340
38
341 37
21
0.4013
9.9588
51
41
97
50
97
50
151
10
151
10
157
58
339
45
340 43
22
0.3479
9.9545
54 22
98
19
98
19
149
41
149
41
156
45
338
52
339 49
23
0.2976
9.9499
57
5
98
49
98
49
148
11
148
11
155
29
338
0
338 54
24
0.2501
9.9451
59
51
99
20
99
20
146
40
146
40
154
11
337
9
338 0
25
0.2053
9.9400
62
40
99
53
99
53
145
7
145
7
152
50
336
18
337 6
26
0.1631
9.9345
65
33
100
28
100
28
143
32
143
32
151
25
335
28
336 13
27
0.1232
9.9287
68
30
101
5
101
5
141
55
141
55
149
56
334
38
335 19
28
0.0857
9.9226
71
33
101
45
101
45
140
15
140
15
148
22
333
49
334 25
29
0.0503
9.9161
74
41
102
27
102
27
138
33
138
33
146
42
333
0
333 32
30
0.0170
9.9092
77
57
103
13
103
13
136
46
136
46
144 55
332
12
332 39
31
9.9857
9.9019
81
23
104
4
104
4
134 56
134
56
142
59
331
24
331 46
32
9.9565
9.8940
85
0
105
1
105
1
132
59
132
59
140
51
330
36
330 54
33
9.9292
9.8856
88
54
106
6
106
6
130
54
130
54
138
27
329
49
330 2
34
9.9040
9.8765
93
11
107
22
107
22
128
38
128
38
135
38
329
2
329 10
35
9.8808
9.8665
98
7
108
58
108
58
126
2
126
2
132
13
328
14
328 19
36
9.8600
9.8555
104
20
111
116
13
34
111
13
122
47
122
47
127
29
327
27
327 28
?'
9.8443
9.8443
116 34
116
34
116 34
116 34
116
H
326
45
326 45
?' =
:36°
52'
LI. 64"
sin q' = 0.6
36
TABLE Va.
- ■
A.
Diff.
B.
Diff.
B'.
Diff. 1^
0
- '^.00
—9.60
— o'.ooo
—11
— o'.ooo
—34
1
9.00
9.00
0.011
11
0.034
34
2
17.99
8.98
0.023
12
0.067
33
3
26.95
8.95
0.034
11
0.101
34
4
35.88
8.91
0.045
11
0.134
33
5
— 44.77
—8.87
—0.057
—12
—0.167
—33
6
53.61
8.80
0.068
11
0.200
33
7
62.37
8.73
0.080
12
0.232
32
8
71.07
8.65
0.092
12
0.263
31
9
79.67
8.56
0.104
12
0.294
31
10
— 88.18
—8.46
—0.117
—13
—0.324
—30
11
96.58
8.34
0.129
12
0.353
29
12
104.86
8.22
0.142
13
0.382
29
13
113.01
8.08
0.156
14
0.409
27
14
121.02
7.94
0.169
13
0.436
27
15
—128.88
—7.79
—0.183
—14
—0.461
—25
16
136.59
7^62
0.197
14
0.486
25
17
144.12
7.43
0.211
14
0.509
23
18
151.47
7.27
0.226
15
0.531
22
19
158.63
7.08
0.241
15
0.552
21
20
—165.60
—6.86
—0.256
—15
—0.571
—19
21
172.35
6.65
0.271
15
0.590
19
22
178.89
6.43
0.287
16
0.606
16
23
185.20
6.20
0.303
16
0.622
16
24
191.28
5.96
0.319
16
0.636
14
25
—J 97.11
—5.71
—0.336
—17
—0.648
—12
26
202.69
5.45
0.352
16
0.659
10
27
208.00
5.18
0.369
17
0.668
9
28
213.05
4.91
0.386
17
0.676
7
29
217.81
4.63
0.403
17
0.682
6
30
—222.30
—4.34
—0.419
—16
—0.687
— 4
31
226.48
4.04
0.436
17
0.690
3
32
230.37
3.74
0.453
17
0.692
1
33
233.95
3.42
0.470
17
0.692
0
34
237.21
3.10
0.486
16
0.691
4-2
35
—240.15
—2.78
—0.502
—16
—0.688
+ 4
36
242.76
2.45
0.518
16
0.683
5
37
245.04
2.11
0.534
16
0.677
6
38
246.98
1.77
0.549
15
0.670
8
39
248.57
1.41
0.564
15
0.661
9
40
—249.80
—1.06
—0.578
—14
—0.651
+11
41
250.68
0.70
0.591
13
0.639
12
42
251.20
0.33
0.604
12
0.627
13
TABLE Va
37
X.
A.
Diff.
B.
Diff.
B'.
Diff.
42
—251.20
— 0.33
— 6'.604
— 12
— 6'.627
+13
43
251.34
+ 0.04
0.615
11
0.613
15
44
251.11
^ 0.42
0.626
11
0.597
16
45
250.50
0.80
0.636
10
0.580
17
46
249.51
1.18
0.645
8
0.563
18
47
—248.13
+ 1.57
—0.652
— 7
—0.544
+19
48
246.36
1.96-
0.659
6
0.524
20
49
244.20
2.36
0.664
4
0.503
21
50
241.64
2.76
0.667
3
0.482
22
51
238.68
3.16
0.669
1
0.459
23
52
—235.31
+ 3.57
—0.669
+ 1
—0.436
+23
53
231.54
3.98
0.667
2
0.412
24
54
227.35
4.39
0.664
4
0.387
25
55
222.76
4.80
0.659
6
0.361
26
56
217.75
5.22
0.651
9
0.335
26
57
—212.32
+ 5.64
—0.641
+ 11
—0.309
+26
58
206.47
6.06
0.629
13
0.282
27
59
200.20
6.47
0.615
15
0.255
27
60
193.52
6.90
0.598
18
0.227
28
61
186.40
7.32
0.579
20
0.200
27
62
—178.87
+ 7.74
-0.557
+ 23
—0.172
+28
63
170.91
8.17
0.532
26
0.144
28
64
162.52
8.60
0.504
29
0.116
28
65
153.70
9.03
0.474
32
0.088
28
66
144.46
9.45
0.440
35
0.061
27
67
—134.79
+ 9.88
—0.403
+ 38
—0.033
+28
68
124.69
10.31
0.363
41
—0.006
27
69
114.16
10.74
0.320
45
+0.021
27
70
103.20
11.17
0.273
49
0.048
27
71
91.81
11.60
0.222
52
0.074
26
72
— 80.00
+12.03
—0.168
+ 56
+0.099
+25
73
67.75
12.46
0.110
59
0.124
25
74
55.07
12.89
0.049
63
0.148
24
75
41.97
13.32
+0.016
67
0.172
24
76
28.43
13.72
0.086
71
0.195
22
77
— 14.47
+14.18
+0.159
+ 75
+0.216
+21
78
0.07
14.61
0.237
80
0.237
21
79
+ 14.76
15.04
0.319
84
0.257
20
80
30.02
15.47
0.405
88
0.276
19
81
45.70
15.89
0.496
93
0.294
18
82
+ 61.80
f 16.32
+0.591
+ 97
+0.311
+16
83
78.34
16.76
0.691
102
0.326
15
84
95.32
17.19
0.795
106
0.340
13 '
38
TABLE \c
X.
A.
Diff.
B.
Diff.
B'.
Diff.
8°4
+ 90.32
+17.19
+ 0795
+106
+6'.340
+ 13
85
112.72
17.62
0.904
111
0.352
12
86
130.56
18.06
1.018
116
0.363
10
87
148.84
18.49
1.137
121
0.373
9
88
167.54
18.92
1.261
126
0.381
7
89
+ 186.69
+19.36
+ 1.390
+132
+0.386
+ 5
90
206.27
19.80
1.525
137
0.390
91
226.29
20.24
1.665
142
0.392
1
92
246.75
20.68
1.810
148
0.392
— 1
93
267.65
21.13
1.961
154
0.390
3
94
+ 289.01
+21.58
+ 2.118
+159
+0.385
— 6
95
310.82
22.03
2.280
165
0.378
8
96
333.08
22.49
2.449
171
0.368
11
97
355.80
22.95
2.623
178
0.355
14
98
378.99
23.42
2.805
184
0.339
17
99
+ 402.65
+23.89
+ 2.992
+191
+0.320
— 21
100
426.78
24.37
3.187
198
0.297
25
101
451.40
24.86
3.388
204
0.270
28
102
476.51
25.36
3.596
212
0.240
32
103
502.12
25.86
3.812
220
0.205
37
104
+ 528.24
+26.38
+ 4.036
+227
+0.165
— 42
105
554.88
26.90
4.267
235
0.121
47
106
582.04
27.43
4.506
240
0.071
53
107
609.75
27.99
4.755
250
+0.015
59
108
638.02
28.55
5.012
261
—0.048
66
109
+ 666.85
+29.11
+ 5.278
+271
—0.117
— 72
110
696.27
29.72
5.554
281
0.193
80
111
726.29
30.33
5.841
292
0.278
89
112
756.93
30.96
6.138
302
0.371
98
113
788.21
31.61
6.446
314
0.474
108
114
+ 820.15
+32.28
+ 6.766
+326
—0.587
—119
115
852.77
32.98
7.099
339
0.712
131
116
886.11
33.70
7.445
353
0.849
144
117
920.18
34.45
8.806
368
1.000
158
118
955.02
35.22
8.181
383
1.166
174
119
+ 990.65
+36.05
+ 8.572
+399
—1.348
—191
120
J 027.13
36.91
8.980
417
1.548
209
121
1064.47
37.79
9.407
436
1.767
230
122
1102.71
38.73
9.853
456
2.009
253
123
1141.93
39.71
10.320
478
2.274
278
124
+1182.14
+40.74
+10.809
+501
—2.566
—306
125
1223.41
41.82
11.323
527
2.886
336
126
1265.78
42.96
11.863
554
3.239
370 i
TABLE Va.
39
X.
A.
Diflf.
B.
Diff.
B'.
Diff.
126
+1260.78
+ 42.96
+ ll863
+ 0.554
— 3^239
— 0.370
127
1309.33
44.16
12.431
0.584
3.627
0.408
128
1354.11
45.43
13.031
0.616
4.055
0.449
129
1400.20
46.78
13.663
0.651
4.526
0.496
130
1447.67
48.20
14.333
0.690
5.047
0.547
131
+1496.61
+ 49.72
+ 15.043
+ 0.731
— 5.621
— 0.605
132
1547.11
51.33
15.796
0.777
6.257
0.669
133
1599.28
53.04
16.597
0.827
6.960
0.741
134
1653.20
54.87
17.451
0.883
7.739
0.821
135
1709.02
56.82
18.363
0.945
8.603
0.912
136
+1766.84
+ 58.91
+ 19.341
+ 1.013
— 9.563
— 1.014
137
1826.84
61.15
20.389
1.088
10.631
1.128
138
1889.15
63.55
21.517
1.171
11.820
1.258
139
1953.95
66.14
22.732
1.265
13.148
1.406
140
2021.43
68.92
24.047
1.371
14.633
1.573
141
+2091.79
+ 71.90
+ 25.475
+ 1.490
— 16.295
— 1.765
142
2165.28
75.15
27.027
1.623
18.163
1.984
143
2242.15
78.65
28.722
1.774
20.263
2.234
144
2322.68
82.47
30.576
1.946
22.631
2.523
145
2407.20
86.58
32.615
2.143
25.309
2.856
146
+2496.06
+ 91.16
+ 34.862
+ 2.368
— 28.344
— 3.242
147
2589.66
96.11
37.351
2.626
31.794
3.713
148
2688.45
101.56
40.115
2.924
35.730
4.224
149
2792.96
107.54
43.199
3.272
40.233
4.836
150
2903.74
114.13
46.659
3.677
45.403
5.566
151
+3021.46
+121.43
+ 50.553
+ 4.153
— 51.366
— 6.437
152
3146.88
129.53
54.966
4.717
58.267
7.469
153
3280.84
138.56
59.987
5.385
66.295
8.705
154
3424.37
148.67
' 65.737
6.185
75.677
10.202
155
3578.59
160.01
72.357
7.155
86.700
12.024
156
+3744.88
+172.81
+ 80.042
+ 8.328
— 99.726
— 14.260
157
3924.79
187.33
89.014
9.767
115.221
17.023
158
4120.22
203.89
99.577
11.548
133.773
20.471
159
4333.38
222.87
112.111
13.777
156.174
24.815
160
4566.94
244.78
127.132
16.603
183.404
30.348
161
+4824.14
+244.78
+145.317
+20.209
—216.860
— 37.483
162
5108.93
270.26
167.550
24.869
258.371
46.802
163
5426.19
300.11
195.056
31.062
310.464
59.156
164
5782.01
335.39
229.674
39.353
376.683
75.318
165
6184.14
377.50
273.762
50.636
462.100
98.618
166
+6642.49
+428.33
+330.946
+66.405
—574.089
—130.816
167
7170.07
490.43
406.573
88.993
723.733
177.025
168
7784.18
567.43
508.933
122.256
928.140
246.403
16'J
8508.45
651.086
1214.530
CONSTANTS.
Log.
Attractive force of the Sun, h in terms of radius, 0 .0172021 8.2355814
k in seconds, 3548".18761 3.5500066
Length of the Sidereal Year (Hansen and Olufsen), 365^2563582 2.5625978
Length of the Tropical Year, 1850, 365^2422008 2.5625809
Horizontal equatorial parallax of the Sun (Encke),* 8".5776 0.9333658
Constant of Aberration (Struve), 20".4451 1.3105892
Time required for light to pass from. the Ssin to
the Earth, 497^827 2.6970785
Radius of Circle in Seconds of arc, 206264".806 5.3144251
in Seconds of time, 13750^987 4.1383339
Sin r 0.000004848137 4.6855749
Cu-cumference of Circle in Seconds of arc, 1296000" 6.1126050
in Seconds of time, 86400' 4.9365137
in terms of diameter, 7T 3.14159265 0.4971499
General Precession (Struve) 50".2411 + 0''.0002268^
Obliquity of the ecliptic (Struve and Peters), 23° 27'54".22 - 0.4645 z!— .0000014^
in which t is the number of years after 1800
DaHy precession, 1850, 0".1375837 9.1385669
Modulus of Common Logarithms, M 0.4342945 9.6377843
* The Constants of Parallax, Aberration, etc., are those used in the American
the authority for them may be found by reference to the volume for 1855.
(40)
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